1. The original wording by Saidak can be of interest on a special Saidak-page, not on the page titled "Euclid's theorem".
2. I cannot see why the 7-lines version of the proof should be more accessible for laymen than the short version. For:
2a. The central argument in the proof is described in essentially the same way in both versions: "For any n > 1, n and n + 1 have no common factors; they are
coprime" versus "Because all prime divisors of a natural number n are different from the prime divisors of n+1, . . . ".
2b. The short version doesn't have the "induction for any k".
2c. Nor the superfluous and difficult to read and interprete "example" at the end.
2d. Using N2, N3, ..., but not N1, is confusing.
2e. Why 'at least' in '1806 has at least four different prime factors' ?
Who refutes this arguments? Who has a better proposal for the text of the proof? --
Hesselp (
talk) 21:42, 5 April 2017 (UTC)
Since each natural number (≥2) has at least one prime factor, and two successive numbers n and (n+1) don't have any prime factor in common, the product n×(n+1) has more different prime factors than the number n itself.
This implies that each term in the infinite sequence: 1, 2 (1×2), 6 (2×3), 42 (6×7), 1806 (42×43), (1806×1807), (1806×1807) × (1806×1807 + 1), · · · has more different prime factors than the preceding.
The sequence never ends, so the number of different primes never ceases to increase. --
===Second attempt plus === Improved wording of Saidak's proof; discussion in Talk.
Since each natural number (≥2) has at least one prime factor, and two successive numbers n and (n+1) don't have any factor in common, the product n×(n+1) has more different prime factors than the number n itself. So in the chain of
pronic numbers:
1×2 = 2 {2}, 2×3 = 6 {2, 3}, 6×7 = 42 {2,3, 7}, 42×43 = 1806 {2,3,7, 43}, 1806×1807 = 3263443 {2,3,7,43, 13,139}, · · ·
the number of different primes per term will increase forever. --
I'm pleased to see that Lazard (Febr.14, 2017, line 4) describes the meaning of the word series as an expression of a certain type. Less clear (or better: mysterious) is the remark: "obtained by adding together all terms of the associated sequence"; what could be meant by "adding together"? What kind of action should be performed, by who, on which occasion, to obtain / create an expression of the intended kind?
More remarks on the present text of the article, in this Talk page: 15:14 16 April 2017.
To get things clear, I propose to start this article in about the following way:
I n t r o d u c t i o n
In mathematics (
calculus), the word series is primarily used for
expressions of a certain kind, denoting
numbers (or functions).
Symbolic forms like and or expressing a number as the limit of the partial sums of sequence , are called series expression or shorter series.
Secondly, in a more abstract sense, series is used for a certain kind of representation (of a number or a function), and also for a special type of such a series representation named series expansion (of a function, e.g. Maclaurin series, Fourier series).
And thirdly, series can be synonymous with sequence.
Cauchy defined the word series by "an infinite sequence of real numbers".[source: Cours d'Analyse, p.123, p.2, 1821, 2009]
This use of the word 'series' can be seen as somewhat outdated.
The study of series is a major part of
mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures (such as in
combinatorics), through
generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as
physics,
computer science,
statistics and
finance.
C o n t e n t s
D e f i n i t i o n s, c o m m o n w o r d i n g s
Given a infinite sequence with terms et cetera (or starting with ) for which addition is defined, the sequence
is called the sequence of partial sums of sequence .
Alternative notation: .
Example: The sequence 1, 2, 3, 4, ··· is the sequence of partial sums of sequence 1, 1, 1, 1, ··· ; the sequence 1, 1, 1, 1,··· is the sequence of partial sums of sequence 1, 0, 0, 0,··· ; this can be extended in both directions.
A series is a written expression using mathematical signs, consisting of
- an expression denoting the function that maps a given sequence on the limit of its sequence of partial sums
combined with
- an expression denoting an infinite sequence (with addition and distance defined).
Second meaning The symbolic forms (plusses-bullets form) and (capital-sigma form)
are sometimes used to denote the sequence of partial sums of sequence , instead of the value of its eventual existing limit.
A sequence is called summable iff its sequence of partial sums converges (has a finite limit, named: sum of the sequence).
Convergent / divergent series The combination convergent series shouldn't be interpreted literally, for an expression itself cannot be convergent or divergent. By tradition "Σ is a convergent series" as well as "series Σ converges" are used to express that sequence is summable. Similarly, "Σ is a divergent series" and "series Σ diverges" are used to say that sequence is not summable.
Convergence test for series Again, this traditional wording cannot be taken literally because 'series' is the name of an expression of a certain kind, not the name of a mathematical notion. An alternative is: summability test for sequences.
Absolute convergent series This is the traditional naming for a sequence with summable absolute values of its terms. The alternative absolute summable sequence is not in common use.
Series Σ and sequence are interchangeable in traditional clauses like:
- the sum of series Σ , the terms of series Σ , the (sequence of) partial sums of series Σ ,
the
Cauchy product of series Σ and series Σ
- the series Σ is geometric, arithmetic, harmonic, alternating, non negative, increasing (and more).
There is no standard interpretation for the limit of series Σ .
S e r i e s r e p r e s e n t a t i o n o f n u m b e r s a n d f u n c t i o n s
In some contexts the word 'series' shouldn't be seen as referring to a certain type of written symbolic expressions, but as referring to a special type of representation of numbers (and functions). Namely: defining a (
irrational) number as the limit of the partial sums of a known infinite sequence of (rational or irrational) numbers. And in the case of functions: defining a function as the limit of the partial sums of an infinite sequence of functions (which are seen as 'easier' or more elementary in one way or another than the function represented by the limit).
Examples of the use of the word 'series' in this sense, can be seen in the final sentences of the introduction above, starting with "The study of series is a major part ...".
As comparable with the idea of series representation or infinite sum representation can be seen: the continued fraction representation and the infinite product representation (for numbers and functions).
S e r i e s e x p a n s i o n o f f u n c t i o n s
The combination 'series expansion' is used for a special type of series representation of functions. ('Series expansion of numbers ' is meaningless.)
A series expansion is a representation of a function by means of the infinite sum of a sequence of power functions of increasing degree, in one of its variables. Or functions like (for example) .
The labels Maclaurin series, Taylor series, Fourier series shouldn't be seen as denoting expressions but rather representations of the type series expansion. So Maclaurin series should be understood as Maclaurin expansion, Fourier series as Fourier expansion, et cetera. [Source: WolframMathWorld
series expansion and
Maclaurin series].
P o w e r s e r i e s
"Power series" can be used
- as synonym for "Maclaurin expansion", and
- for a series expression which includes a sequence of power functions with increasing degree.
C a u c h y a s s o u r c e o f c o n f u s i o n
Cauchy, in his 'Cours d'Analyse' (1821) made an important, but quite subtile, distinction between the meaning of 'to converge' and 'being convergent':
- a sequence (French: suite) can converge (both French and English) to a limit, versus
- an infinite sequence of real numbers (named 'série' by Cauchy) having its sequence of partial sums converging to a limit, the first sequence named 'une série convergente ' .
Only a tiny difference between 'sequence' and 'series', but an essential one between 'converging' and 'convergent'.
This imprudent choise caused permanent confusion around the use of the word 'series'(e.g. in the German translations of 'Cours d'Analyse' of 1828 and 1885), until the present day.
[sources:
Cauchy, see p.123 and p.2 quantité
C.L.B. Susler, 1828,
Susler, S.92,
Carl Itzigsohn, 1885,
Bradley/Sandifer, 2009 ]
[More sources on the problem with 'series' in books/publications by: professor H. Von Mangoldt, E.J. Dijksterhuis, H.B.A. Bockwinkel, professor N.G. de Bruijn, professor A.C.M. van Rooij, professor D.A. Quadling, Mike Spivack, H.N. Pot; links have to be added. Several of this sources are written in Dutch.]
Overwegen om delen van de rest van dit artikel onder te brengen op andere pagina's. Vaak staat er nu al een verwijzing naar "Main article".
Rekenregels Rewriting voor serie expressions
To 166.216.158.233, and ... . On Februari 28 2017, you changed {sk} into (sk) at several places. I understand your argument (a sequence is a mapping, not a set), but I see your solution as insufficient. For without any harm, you can do without braces/parentheses at all, and without any index symbol as well. A sequence is defined as a mapping on the set of naturals, so label them with a single letter. Just as people mostly do with mappings/functions with other sets as domain: f, g, F, G, ... .
When there is a risk of confusion you can write "sequence s", "sequence S" in stead of just "s" or "S".
Who has objections? (Yes, I know the index is tradition, but it is superfluous and therefore disturbing.)
In the Definition section, three lines after "More generally ..." I read:
the function is a sequence denoted by .
I count three different notations for the same domain--function (sequence), four lines later a fourth version - - is used.
Last remark: It's not correct to say that sequences ( and ) are subsets of semigroup . --
For me it is impossible to find any information in the second part of subsection 'Definition'- after 'More generally....'.
The text seems to suggest that the notion of "series" (whatever that is ...) can be extended from something associated with sequences (mappings on the set of naturals) to a comparable 'something' associated with mappings on more general index sets. But nothing is said about how such generalized mappings can be transformed into a limit number . Is it possible to generalize the tric with the 'partial sums'? This index sets has to be countable? No reference is given. (The present text is composed by Chetrasho
July 27, 2011).
I propose to skip the text from 'More generally' until 'Convergent series'. Any objections? --
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My editing priorities: 1.Correct and non-misleading statements 2.Sufficient referencing 3.Accessible leads 4.Article structure
Hello Bill Cherowitzo. You are right, my two questions are answered in the final section of the article.
But I persist that the description of the notion named series becomes even more unclear by adding six sentences (the greater part of the Definition section) on a generalization that will be unknown to most readers.
Moreover, the correlation between the position of this notion connected with sequences, and its position connected with mappings on an index set, is not very strong. For:
In (elementary) calculus two different symbolic forms (both named 'series') are used, expressing the relation between a sequence and its 'sum'. One of them, the plusses-bullets form cannot be used in the generalized situation. And the other one, the capital-sigma form needs adaption ( instead of or or or ).
The absence of relevant information in this six sentences is not undone by a 'lack of information tag'. Skipping this sentences I cannot see as a "removal of [relevant] material". --
Openingszin datum ???
Infinite series
The sum of an infinite series a0+a1+a2+... is the limit of the sequence of partial sums
S N = a 0 + a 1 + a 2 + . . . + a N {\displaystyle S_{N}=a_{0}+a_{1}+a_{2}+...+a_{N}} {\displaystyle S_{N}=a_{0}+a_{1}+a_{2}+...+a_{N}} ,
as N → ∞. This limit can have a finite value; if it does, the series is said to converge; if it does not, it is said to diverge. The fact that infinite series can converge resolves several of Zeno's paradoxes.
Openingszin 30 March 2006 16:48
In
mathematics, a series is often represented as the
sum of a
sequence of
terms. That is, a series is represented as a list of numbers with
addition operations between them, e.g,
which may or may not be meaningful, as it will be explained below.
Yesterday's (13 April 2017) reduction in this section is an improvement, yes. Now this shorter version makes it easier to explain my objection to its central message. I paraphrase this message in the next four lines:
1. For any sequence is defined a
2. associated series Σ (defined as: an ordered "element of the free abelian group with a given set as basis" - the link says).
3. To series Σ is associated the
4. sequence of the partial sums of .
Why in line 2 an 3 a detour via a double 'association'(?) with something named 'series'? Is the meaning of that word clearly explained in this way to a reader? I don't think so. I'm working on a text that starts with:
"In mathematics the word series is primarily used for expressions of a certain kind, denoting numbers (or functions). Secondly"
I plan to post this within a few days. --
Hesselp (
talk) 13:39, 14 April 2017 (UTC).
Who can tell me how to find out whether or not a given "ordered element of the free abelian group with a given set as basis" has 100 as its sum? Who can mention a 'reliable source' where the answer can be found?
Why should this mysterious serieses be introduced at all, in a situation where it's completely clear what it means that a given sequence has 100 as its sum. I cannot find a motivation for this in a 'reliable source' mentioned in the present article.
So skip this humbug (excusez le mot).
About an eventual 'immediate removal': Should I have to expect that a majority in the Wiki community will support removing a serious attempt to describe in which way (ways!) the word 'series' is used in most existing mathematical texts. And replace a version including a 'definition' which has nothing to do with the way this word is used in practice; only because the wording has some resemblance with meaningless wordings that can be found in (yes, quite a lot of) textbooks.
In the present 'definition' of series the words 'formal sum' are linked to a text on Free abelian groups. Can this be seen as a 'reliable source' for a reader who wants to know what could be meant by 'formal sum'? Wikipedia is not open for attempts to improve this? --
Line 3, quotation: "Summation notation....to denote a series, ..."
A notation to denote an expression ?? Sounds strange (first sentence says: series = expression of certain kind).
Line 4, quotation: "Series are formal sums, meaning... by plus signs),"
I can read this as: "The word 'sum' has different meanings, but the combination 'formal sum' is a substitute for 'series' (being forms consisting of sequence elements/terms separated by plus signs)". Correct?, this is what is meant?
But "Series are formal sums" seems to communicate not exactly the same as " 'series' is synonym with 'formal sum' ".
Line 4-bis, quotation: "these objects are defined in terms of their form"
With 'these objects' will be meant: 'these expressions (as shown in the first sentence)', I suppose. But then I miss the sense of this clause. An expression IS a form, and don't has to be defined (or described?) in terms OF its form.
Line 6. Properties of expressions? and operations defined on expressions? This regards operations as enlarging, or changing into bold face, or ...?
Line 7. "...convergence of a series". In other words: "convergence of a certain expression"? I'm lost.
I'll show an alternative. --
=== Three proposals for adaptations in the Definition section ===
I. Note 3 in the present text, saying "...a more abstract definition....is given in....", should be removed.
For it doesn't have any sense to refer to a 'more abstract definition' of
an expression of the form , labeled with the name 'series'.
There is not a 'less abstract definition' of this kind of expression either. Only a description.
II. A more direct formulation of the third sentence in this section is:
"A series is also called formal sum, for a series expression has a well-defined form with plus signs."
III. Remarks on the 'usefullness' and the 'fundamental propery' of such expressions of the form , shouldn't be included in a definition section. --
H o w t o r e d u c e c o n f u s i o n
The best thing to do is: Stop using the word 'series' at all, and say:
(absolute) summable sequence and summability tests, in stead of: (absolute) convergent series and convergence tests .
Second best is: inform students and readers of Wikipedia about the historical source of the confusion. Let them understand that the existence of any definable notion 'series' (different from 'sequence') is a wide-spread misconception. And train them to interprete (absolute) convergent series as nothing else as summable sequence. --
Answer to Lazard: Thank you for contributing to the search for the best way to describe what is meant with the word 'series' in texts on mathematics(calculus). I saw some points in your rewriting of the Definition section which I can see as improvements. But there are some problems left:
1. Rewording the first sentence more close to the usual way as definition of 'infinite series / series', I get:
An infinite sum is called series or infinite series if represented by an expression of the form: where ...
This paraphrasing is correct?
Please add an explanation of what you mean by 'infinite sum'. &nbnp;And tell how a blind person can decide whether or not he is allowed to say 'series' to such an infinite sum, as he cannot see the form of the representation.
2. In the third sentence 'summation notation' is introduced, showing a 'capital-sigma' form, followed by an equal sign and a 'plusses-bullets' form. Why two different forms to illustrate the 'summation notation'?
3. Please explain what you mean with 'formal sum' (fourth sentence). See
this discussion. And the same question for 'summation' at the end of that sentence.
4. Your seventh sentence end with "...the convergence of a series". Do you really mean to define "the convergence of an expression(of a certain type)?
5. Finally, I'ld like to see an explanation of the clause "the expression obtained by adding all those
[an infinite number of] terms together" (fifth sentence in the intro). I don't see how the activity of 'adding' (of infinite many terms!) can have an 'expression' as result. --
A sum of two numbers given in series representation,
a product of two numbers given in series representation, and
a product of two numbers, one of them given in series representation,
can be reduced according to:
The same applies for functions instead of numbers. --
Invoeren van mijn alternatief, tot aan "Examples", te wijzigen is "Examples of the use of the series representation', of 'series expressions', and of the word 'series' in different situations" .
Adjective in combination with: 'expression', 'representation', expansion'.
Noun as - a contraction of 'series expression'
-as synonym with 'seqence',
- as part of traditional wordings: (absolute) convergent series, convergence test for series, Cauchy product of two series, meaning ............
Jeder Folge ist identisch mit der Partialsummenfolge seiner Differenzenfolge, also: 'Reihe' und 'Folge' sind synonym.
Deswegen sagt Satz 4 der Definition:
"Falls die Folge/Reihe konvergiert, so nennt man die Grenzwert der Folge/Reihe auch Summe der Folge/Reihe " .
Korrekt? Was ist hier definiert?
Zur Zeit wird eine Alternative diskutiert im 'Talk page' der englische
Wikipedia.
--
Hej/Hallo. Hier versucht ein Holländer auf Deutsch zu schreiben.
Ich (*1942) bin schon sehr lange interessiert in die Frage um das Unterschied (falls existierend) zwischen 'Folge' und 'Reihe'. Heute entdeckte ich diese Wiki-Seite, wo ich meine Frage sehr ausführlich behandelt sehe. Aber am Ende lese ich doch wieder im Definition (Reihe):
'Reihe' ist die Nahme der Schreibweise (in Symbolform) für (in Textform) "die Partialsummenfolge der Folge ".
Folglich lese ich im Definition (Grenzwert einer Reihe):
Die Grenzwert einer Schreibweise , ist der Limes ......
Meine ewige Frage bleibt: Wie kann eine Schreibweise (englisch: a symbolic expression) einen Grenzwert haben? (oder: konvergent sein, oder eine Summe haben, oder konvergieren, oder ...). Das ist doch Unsinn?
Für meine Versuche die Sache auf zu klären, sehe
Ganz klar ? oder
An attempt to clarify the 'series' mystery
--
Hesselp 12:05, 19. Apr. 2017 (CEST)
Die Definition von 'Reihe' im heutigen Text lautet:
Für eine reelle Folge ist die Reihe die Folge aller Partialsummen .
Hier wird (meiner Meinung) nichts anderes gesagt als:
Mit anderen Worten:
ist eine (kürzere) Schreibweise für die Partialsummenfolge einer reelle Folge ;
ein solcher Ausdruck - die Sigma-Schreibweise für Folgen - wird Reiheform oder Reihe genannt.
Korte titel: Paraphrasierung der Reihe-Definition.
Verdediging bij skippen: Es soll zumindestens erklärt werden warum die Paraphasierung nicht richtig ist.
Bij reactie "overbodig": oude regel vervángen door de nieuwe. De oude is impliciet en daarom erg onduidelijk.
Stephan Kulla Reaktion zu "sollte anstelle von einer Reihe, von einer Folge sprechen?
Tja, eigentlich: ja!. Oder....obwohl...historisch gesehen sind die Wörte 'Reihe' und 'Folge' sehr oft wie Synonyme gebraucht (Und auch heute noch: sehe z.B.
[1] "Folge und Reihe sind also nicht scharf voneinander trennbar. Die
Zeitreihen der Wirtschaftswissenschaftler sind eigentlich Folgen."
Und sehe
Esperanto: Rimarko: Ne ekzistas formala diferenco inter la nocioj
vico kaj serio. )
Darum kann man auch vorschlagen beide Wörte durcheinander zu benutzen.
Nochmals:
Die 'Folge der Partialsummen einer Folge' ist wieder eine Folge.
Der Limit einer 'Folge der Partialsummen einer Folge' ist ein Zahl.
E s g i b t k e i n m a t h e m a t i s c h e B e g r i f f d a z w i s c h e n / d a n e b e n .
Sehr viele Autoren schreiben und sprechen als ob es ein solcher Zwischenbegriff (ich sagte: 'Gespenst') gibt. Aber wenn man genau analysiert wie das 'Zwischenbegriff' (vielmals 'Reihe' genannt) definiert wird, dann kommt man nicht weiter als: "die Sigma-Schreibweise für Folgen". (Und dazu meistens auch: die "plusses/bullets notation" a1+a2+a3+··· . (Plusse/Punkte-Schreibweise ?))
Reaktion zu: "Ich verstehe das Problem noch nicht."
Das Wort 'Reihe' wird definiert als die Nahme einer symbolisch geschrieben mathematische Ausdruck (die Sigma-Schreibweise). Aber überall im Text wird gesprochen von 'konvergierende Reihe', 'konvergente Reihe', 'Summe der Reihe', 'Partialsummen der Reihe', usw. Wobei nirgends gesagt wird das der Wikibook-Leser die Schreibweise-Definition vergessen muss ('Reihe' wird niemals in dieser Auffassung gebraucht), und das traditionelle 'Summe der Reihe' muss lesen wie 'Summe der Folge'. Und 'konvergente Reihe' wie 'summierbare Folge'. Es kann dabei darauf hingewesen werden das Cauchy sehr unvorsichtig 'convergente' wählte als adjective für eine Zahlenfolge mit konvergierende Partialsummen (eine summierbare Zahlenfolge).
--
Part of a series of articles about |
Calculus |
---|
In mathematics (
calculus), the word series is primarily used as adjective specifying a certain kind of
expressions denoting
numbers (or functions).
Symbolic forms like and or expressing a number as the limit of the partial sums of sequence , are called series expression. 'Series expression' is often shortened to just 'series'.
Secondly, series is used as an adjective in series representation, denoting the kind of representation (of a number or a function) as a limit of the partial sums of a given sequence.
Thirdly, series is used, again as an adjective, in series expansion. Being a special type of series representation (of functions, not numbers). For instance:
the Maclaurin expansion of a given function and the Fourier expansion of a given function are series expansions.
Finally, (the noun) series can be synonymous with sequence. Cauchy defined the word series by "an infinite sequence of real numbers". [1] The use of the word 'series' for 'sequence' has a long tradition, with analogons in other languages, but seems to be considered as somewhat outdated.
The rather widespread idea about the existence of a mathematical notion (a definable mathematical object, called 'series'), 'associated' in some way with a given number sequence, with its partial sums sequence, and with the eventual limit thereof, is false.
The study of the series representation is a major part of
mathematical analysis. With this tool,
irrationals can be described/defined by means of (the limit of) a relatively easy descriptable sequence of rationals.
This kind of representation is used in most areas of mathematics, even for studying finite structures (such as in
combinatorics), through
generating functions. In addition to their ubiquity in mathematics, the series representation is also widely used in other quantitative disciplines such as
physics,
computer science,
statistics and
finance.
Given a infinite sequence with terms et cetera (or starting with ) for which addition is defined, the sequence
is called the sequence of partial sums of sequence .
Alternative notation: . Alternative name: the sum sequence of (sequence)
[2].
Example: The sequence (1, 2, 3, 4, ···) is the sum sequence of (1, 1, 1, 1, ··· ); being the sum sequence of (1, 0, 0, 0, ··· ); this can be extended in both directions.
A series, short for series expression, is a written expression using mathematical signs, consisting of
- an expression denoting the function that maps a given sequence on the limit of its sum sequence, combined with
- an expression denoting an infinite sequence (with addition and distance defined).
Examples: (plusses-bullets notation), (capital-sigma notation).
Sometimes, the same symbolic forms are used to denote the sum sequence of , instead of the value of its eventual limit.
A sequence with a converging sum sequence is called summable. The finite limit is called sum of the sequence.
A valid series expression has a summable sequence as its argument (and denotes a value). Otherwise the expression is void. Traditional wordings are: "convergent/divergent series expression" or "convergent/divergent series".
Convergent / divergent series The combination convergent series shouldn't be interpreted literally, for an expression itself cannot be convergent or divergent. By tradition "Σ is a convergent series" as well as "series Σ converges" are used to express that is summable. Similarly, "Σ is a divergent series" and "series Σ diverges" are used to say that is not summable.
Convergence test for series Again, this traditional wording cannot be taken literally because 'series' is the name of an expression of a certain kind, not the name of a mathematical notion. An alternative is: summability test for sequences.
Absolute convergent series This is the traditional naming for a sequence with summable absolute values of its terms. The alternative absolute summable sequence is not in common use.
Series Σ and sequence are interchangeable in traditional clauses like:
- the sum of series Σ , the terms of series Σ , the (sequence of) partial sums of series Σ , the
Cauchy product of series Σ and series Σ
- the series Σ is geometric, arithmetic, harmonic, alternating, non negative, increasing (and more).
There is no standard interpretation for the limit of series Σ .
In some contexts the word 'series' shouldn't be seen as referring to a certain type of written symbolic expressions, but as referring to a special type of representation of numbers (and functions). Namely: defining a (
irrational) number as the limit of the partial sums of a known infinite sequence of (rational or irrational) numbers. And in the case of functions: defining a function as the limit of the partial sums of an infinite sequence of functions (which are seen as 'easier' or more elementary in one way or another than the function represented by the limit).
As comparable with the idea of series representation (or: infinite sum representation) can be seen: the continued fraction representation and the infinite product representation (for numbers and functions).
R e d u c t i o n o f s u m s a n d p r o d u c t s
A sum of two numbers given in series representation,
a product of two numbers given in series representation, and
a product of two numbers, one of them given in series representation,
can be reduced according to:
(sequence or sequence summable)
.
The same applies for functions instead of numbers.
The name
'series expansion' is used for a special type of series representation of functions. (Not applicable to numbers.)
A series expansion is a series representation of a function, using a sequence of power functions of increasing degree, in one of its variables. Or functions like (for example) .
The labels Maclaurin series, Taylor series, Fourier series shouldn't be seen as denoting expressions but rather representations of the type series expansion. So Maclaurin series should be understood as Maclaurin expansion, Fourier series as Fourier expansion, et cetera.
[3]
The name
power series can occur
- as synonym for Maclaurin expansion, and
- denoting a series expression which includes an expression for a sequence of power functions with increasing degree.
Cauchy, in his 'Cours d'Analyse' (1821) made an important, but quite subtile, distinction between the meaning of 'to converge' and 'being convergent':
- A sequence (French: suite) can converge to a limit.
- A sequence with converging partial sums, is called convergent by Cauchy (meaning 'summable')
Moreover, an infinite sequence with real numbers as terms, he called a series (French: série).
This imprudent choise caused permanent confusion around the use of the word 'series' (e.g. in the German translations of 'Cours d'Analyse' of 1828 and 1885)
[4] until the present day.
Below, the words 'series' and 'convergent / divergent' are not always used conform the preceding descriptions. In such cases the context has to be taken into account to track down the intended meaning.
[More sources on the problem with 'series' in books/publications by: professor H. Von Mangoldt, E.J. Dijksterhuis, H.B.A. Bockwinkel, professor N.G. de Bruijn, professor A.C.M. van Rooij, professor D.A. Quadling, Mike Spivack, H.N. Pot; links have to be added. Several of this sources are written in Dutch.] --
Hesselp (
talk) 15:38, 16 April 2017 (UTC)
H o w t o r e d u c e c o n f u s i o n
The best thing to do is: Stop using the word 'series' at all, and say:
(absolute) summable sequence and summability tests, in stead of: (absolute) convergent series and convergence tests .
Second best is: inform students and readers of Wikipedia about the historical source of the confusion. Let them understand that the existence of any definable notion 'series' (different from 'sequence') is a wide-spread misconception. And train them to interprete (absolute) convergent series as nothing else as (absolute) summable sequence. --
Hesselp (
talk) 13:12, 17 April 2017 (UTC)
The present text strongly suggests that there is only one correct interpretation of what is meant by the word 'series' in mathematical texts. That is that the word 'series' is the name for a certain idea / notion / conception / entity. But what IS "it"?
"It" is NOT a number.
"It" is NOT a sequence (= a mapping on N)
"It" is NOT an expression (for the present text says: "a series is represented by an expression)
"It" is NOT a function.
"It" is 'associated' (what's that?) with a sequence. "It" is sometimes 'associated' with a value.
"It" has terms and partial sums.
"It" can have a limit, a value, a sum.
"It" can be geometric, arithmetic, harmonic, alterating, converging, diverging, absolute converging, and more.
What's in fact the content of this black "it"-box? It seems to be empty.
I'm going to replace this unsatisfactory text by an alternative introduction. Chiefly identical with what was shown in this Talk page
here, 18 April 2017. The only reaction on it was the remark that "Hesselp doesn't understand what A SERIES IS (in mathematics)". I agree with that. --
It seems like you have identified a Dutch school of thought on this topic. This would probably be good for a paragraph in the article, but certainly not a rewrite.--Bill Cherowitzo (talk) 05:20, 27 April 2017 (UTC)
About: expressing a number or a function by means of an infinite series. See:
[2]
The present text presents in the intro plus subsection Definition, four different 'definitions', all of them using the wording:
"a series IS ..." .
1. (Intro, sentence 1) "a series IS ... the sum of the terms of ..."
(Being the sum of numbers again a number, the words 'series' and 'number' are synonym.)
2. (Intro, sent.5) "The series of (associated with) a given sequence a IS the expression a1+a2+a3+··· "
(The word 'series' used as the name of a mapping.)
3. (Definition, sent.1) "a series IS an infinite sum, which is represented by a written symbolic expression of a certain type."
(It isn't clear whether or not the clause after the comma is part of the definition. 'IS' a series still an infinite sum, in situations where it is not represented by an expression of the named form?)
4. (Definition, sent.6) "series(pl) ARE elements of a total algebra of a ring over the monoid of natural numbers over the a commutative ring of the a's "
(The word 'series' as the name for elements of a certain structure; just as the word 'number' is used as the name for elements of another mathematical structure. To which element in this 'definition' is referred by the a's ? )
In case it is true, that the word 'series' has four different meanings in mathematics (is used in four different ways) the article headed by "Series" should be structured like:
a. The word 'series' is used as name/label for ......... .
b. The word 'series' is also used as name/label for ......... .
c. The word 'series' is used as name/label for .......... as well.
d. Moreover, sometimes the word 'series' is used as name/label for ......... .
The present text directs the reader to believe that there is ONE and only ONE sacred given-by-God-meaning of this word.
That's religion, not mathematics.
Do you think, Wcherowi, the summing up of different meanings is wrong?
Do you think, D.Lazard, the summing up of different meanings is wrong?
Do you think, MrOllie, the summing up of different meanings is wrong?
Do you think, Sławomir Biały, the summing up of different meanings is wrong?
One of the main reasons I see the present text as ready for improvement, I described earlier in
"It" is NOT a number.
"It" is NOT a sequence (a mapping on N)
"It" is NOT an expression (for the present text says: "a series is represented by an expression)
"It" is NOT a function.
"It" is 'associated' (what's that?) with a sequence. "It" is sometimes 'associated' with a value.
"It" has terms and partial sums.
"It" can have a limit, a value, a sum.
"It" can be geometric, arithmetic, harmonic, alterating, converging, diverging, absolute converging, and more.
What's in fact the content of this black "it"-box? It seems to be empty.
I'm going to replace this unsatisfactory text by an alternative introduction. Chiefly identical with what was shown in this Talk page
here, 18 April 2017. The only reaction on it was the remark that "Hesselp doesn't understand what A SERIES IS (in mathematics)". I agree with that. --
Hesselp (
talk) 22:05, 24 April 2017 (UTC)
--
1. (Sent.1) "a series IS ... the sum of the terms of ..."
Being the sum of numbers again a number, the words 'series' and 'number' are used as synonym. A few lines later it is said that this is not intended.
2. (Sent.2) "a series continues indefinitely"
What is meant by: an indefinitely continuing 'sum of the terms of something' ?
3. (Sent.4) "the value of a series"
What is meant by: the value of a sum (a number) ?
4. (Sent.4) "evaluation of a limit of something"
What's meant with this?
Is it true that a series doesn't have a value, without that limit being 'evaluated' ?
Is it always possible to 'evaluate' the limit of a sequence of terms ?
5. (Sent.5) "the expression obtained by adding all those (an infinite number of) terms together"
A (symbolic, written) expression can be obtained by writing down some symbols using a pen or pencil (or using the keys of a keyboard). The task of adding an infinite number of terms is not feasible, so never any expression will be obtained.
6. (Sent.6) "obtained by placing the terms side-by-side with pluses in between them.
This 'placing' sounds much better feasible. I miss the three centered dots ('bullets') at the right end.
7. (Sent.6) "infinite expression"
I see 'series' and 'infinite sum' used as synonyms for 'infinite expression'. But what notion / mathematical object is denoted by this labels ? It must be a notion 'not being a part of the conventional foundations of mathematics'. How many readers of this article are acquainted with this notion already by themselves?
8. (Sent.7) "The infinite expression can be denoted ..." Such expressions mostly denote a number, a function or a sequence. But an expression denoting a expression sound very strange.
9. (Sent.9) "two series of the same type"
I cannot find where is explained what is meant by: 'the type of that mysterious notion called series '.
10. (Sent. 8, 9, 10, 11, 12)
Is the (intended) information communicated by this five sentences really of enough importance to be incorporated in the 'introduction' ?
11. (First line after 'Definition') The twofold description of the meaning of the word 'series' (as sum, and as expression) causes - unnecessary? - complexity.
--
@Slawomir. Never in my life I've denied that mathematical expressions are totally different from numbers. You must have misunderstood me somewhere, I cannot trace back where this could have happened.
I agree with you on everything you wrote in the first 7 sentences in 12:46, 2 May 2017(UTC) (Until "The sigma notation for a series..."). About your sentences 8, 9, 10 I'm not sure.
Maybe things become more clear from your judgment of the following statements a - h (true or false):
a) the expression e+π evaluates to (= has as its value) the number e+π
b) the expression 1+1 evaluates to the number 1+1
c) the expression 1+1 evaluates to the number 2
d) the sigma expression Σi =1∞ ai evaluates to the infinite expression a1+a2+a3+···
e) Provided that limn→∞ (a1+ ··· +an) exists,
in other words limn→∞ (a1+ ··· +an) is a valid expression,
in other words sequence (an) is summable,
the infinite expression a1+a2+a3+··· (number-interpretation) evaluates to the number limn→∞ (a1+ ··· +an)
f) the infinite expression a1+a2+a3+··· (sequence-interpretation) evaluates to the sequence (a1+ ··· +an)n≥1
g) Being p1, p2, p3, ··· successive primes,
the infinite expression p1-3+ p2-3 + p3-3+ ··· evaluates to the number p1-3+ p2-3 + p3-3+ ···
h) the infinite expression 9− 9^1+ 9− 9^2+ 9− 9^3+ ··· evaluates to the number Σi =1∞ 9− 9^í
According to me this is a quite peculiar way to use the verb 'to evaluate' (in the intro of the present text: "A series is thus evaluated by examining ...."); you can show sources? I only saw it, meaning: given an expression (denoting a number), find the decimal representation of its value, exact or approximated. --
The intro of the present text explains the meaning of "series" using:
The series of an given infinite sequence is the infinite expression that is obtained by placing terms side-by-side with pluses in between.
By 'infinite expression' is not meant an expression with infinite physical dimensions. Nor an expression of the type "1/0".
The
Wikipedia article says: "an expression in which some operators take an infinite number of arguments". That's sufficiently clear to most of our readers? I doubt
Moreover, that article has: "Examples of well-defined infinite expressions include infinite sums, whether expressed using summation notation or as an infinite series, ....". With a circulating reasoning, because 'infinite sum' is linked to the article named ....'Series (mathematics)'. --
In the discussion on the concept/object/idea/entity (mathematical or philosophical) named "series", a number of negative statements are made on this Talk page, since April 10, 2017. Two new ones are found in a post by Sławomir Biały,
dated 21:57, 2 May 2017(UTC): - "it" is NOT a numeral - "it" is NOT an expression that denotes a number.
"It" is NOT a sequence (a mapping on N)
"It" is NOT an expression
"It" is NOT a function
"It" is NOT a part of Zermelo-Fraenkel set theory
"It" is NOT a part of the conventional foundations of mathematics.
"It" is NOT a numeral
"It" is NOT an expression that denotes a number
"It" is 'associated' (what's that?) with a sequence.
"It" has terms and partial sums.
"It" can have a limit, a value, a sum.
"It" is represented by an expression
"It" is sometimes 'associated' with a value.
"It" can be geometric, arithmetic, harmonic, alterating, converging, diverging, absolute converging, and more.
--
Bovenstaande is (nog) niet geplaatst!
On edit 15:34, 4 May 2017.
What is meant:
a series is a description of the operation: adding one-by-one infinitely many quantities
or
a series is the operation: adding one-by-one infinitely many terms ?
What a reader should think of: an operation that cannot be carried on (not 'effectively') ?
I'm curious to see how you define (based on reliable sources): "a convergent infinite adding operation", "a alternating infinite adding operation" "a geometric infinite adding operation" "a Fourier infinite adding operation" "the Cauchy product of two infinite adding operations" "a power infinite adding operation" and much more.
Please present a mature proposal for the intro-plus-definition part of the article. Here on Talk page, so not unnecessary disturbing our Wiki-readers . --
In Talk page, no user took part in discussion on the merits of the content of this text. So 'no consensus' cannot be a valid reason to revert.
From the 'edit summary' 13:22, 5 May 2017: "...most editors have already given up trying to communicate with you" .
That 'trying to communicate' refers to reactions with no more relation to the content of the proposed text-section, than in phrases of the type:
- don't agree with proposed changes - undocumented POV-pushing - Hesslp doesn't understand what a series is - this talk page is not for discussing personal opinions about the practice of mathematicians - this is not mathematics, it is philosophy - you have clearly a misconception of what is mathematics - for being clearer, every line of Hesselp's post is either wrong, or does not belong to this talk page or both - I reiterate my objection .
Attempts made to start discussion, in this list:
"It" is NOT a number.
"It" is NOT a sequence ( a mapping on N)
"It" is NOT an expression
"It" is NOT a function.
"It" is NOT a part of Zermelo-Fraenkel set theory
"It" is NOT an expression that denotes a number
"It" is NOT a numeral
"It" is represented by an expression
"It" is 'associated' (what's that?) with a sequence.
"It" is sometimes 'associated' with a value.
"It" has terms and partial sums.
"It" can have a limit, a value, a sum.
"It" can be geometric, arithmetic, harmonic, alterating, converging, diverging, absolute converging, and more.
1. (Intro, sentence 1) "a series IS ... the sum of the terms of ..."
(Being the sum of numbers again a number, the words 'series' and 'number' are used as synonym.)
2. (Intro, sent.5) "The series of (associated with) a given sequence a IS the expression a1+a2+a3+··· "
(The word 'series' used as the name of a mapping.)
3. (Definition, sent.1) "a series IS an infinite sum, which is represented by a written symbolic expression of a certain type."
(It isn't clear whether or not the clause after the comma is part of the definition. 'IS' a series still an infinite sum, in situations where it is not represented by an expression of the intended form?)
4. (Definition, sent.6) "series(pl) ARE elements of a total algebra of a ring over the monoid of natural numbers over the a commutative ring of the a's "
(The word 'series' as the name for elements of a certain structure; just as the word 'number' is used as the name for elements of another mathematical structure. To which element in this 'definition' is referred by "the a's" ? )
In case it is accepted that the word 'series' has four different meanings in mathematics (is used in four different ways) the first part of the article headed by "Series" should be structured like:
a. The word 'series' is used as name/label for ......... .
b. The word 'series' is also used as name/label for ......... .
c. The word 'series' is used as name/label for .......... as well.
d. Moreover, sometimes the word 'series' is used as name/label for ......... .
The present text directs the reader to believe that there is ONE and only ONE sacred given-by-God-meaning of this word.
That's religion, not mathematics.
Do you think, Wcherowi, the summing up of different meanings is wrong?
Do you think, D.Lazard, the summing up of different meanings is wrong?
Do you think, MrOllie, the summing up of different meanings is wrong?
Do you think, Sławomir Biały, the summing up of different meanings is wrong?
Being the sum of numbers again a number, the words 'series' and 'number' are used as synonym. A few lines later it is said that this is not intended.
2. (Sent.2) "a series continues indefinitely"
What is meant by: an indefinitely continuing 'sum of the terms of something' ?
3. (Sent.4) "the value of a series"
What is meant by: the value of a sum (a number) ?
4. (Sent.4) "evaluation of a limit of something"
What's meant with this?
Is it true that a series doesn't have a value, without that limit being 'evaluated' ?
Is it always possible to 'evaluate' the limit of a sequence of terms ?
5. (Sent.5) "the expression obtained by adding all those (an infinite number of) terms together"
A (symbolic, written) expression can be obtained by writing down some symbols using a pen or pencil (or using the keys of a keyboard). The task of adding an infinite number of terms is not feasible, so never any expression will be obtained.
6. (Sent.6) "obtained by placing the terms side-by-side with pluses in between them.
This 'placing' sounds much better feasible. I miss the three centered dots ('bullets') at the right end.
7. (Sent.6) "infinite expression"
I see 'series' and 'infinite sum' used as synonyms for 'infinite expression'. But what notion / mathematical object is denoted by this labels ? It must be a notion 'not being a part of the conventional foundations of mathematics'. How many readers of this article are acquainted with this notion already by themselves?
8. (Sent.7) "The infinite expression can be denoted ..." Such expressions mostly denote a number, a function or a sequence. But an expression denoting a expression sound very strange.
9. (Sent.9) "two series of the same type"
I cannot find where is explained what is meant by: 'the type of that mysterious notion called series '.
10. (Sent. 8, 9, 10, 11, 12)
Is the (intended) information communicated by this five sentences really of enough importance to be incorporated in the 'introduction' ?
I agree with you on everything you wrote in the first seven sentences in 12:46, 2 May 2017(UTC) (Until "The sigma notation for a series..."). About your sentences 8, 9, 10 I'm not sure. Maybe things become more clear from your judgment of the following statements a - h (true or false, or ...):
a) the expression e+π evaluates to (has as its value) the number e+π
b) the expression 1+1 evaluates to the number 1+1
c) the expression 1+1 evaluates to the number 2
d) the sigma expression Σi 1∞ ai evaluates to the infinite expression a1+a2+a3+···
e) Provided that limn→∞ (a1+ ··· +an) exists,
in other words limn→∞ (a1+ ··· +an) is a valid expression,
in other words sequence (an) is summable,
the infinite expression a1+a2+a3+··· (number-interpretation) evaluates to the number limn→∞ (a1+ ··· +an)
f) the infinite expression a1+a2+a3+··· (sequence-interpretation) evaluates to the sequence (a1+ ··· +an)n≥1
g) Being p1, p2, p3, ··· successive primes,
the infinite expression p1-3+ p2-3 + p3-3+ ··· evaluates to the number p1-3+ p2-3 + p3-3+ ···
h) the infinite expression 9− 9^1+ 9− 9^2+ 9− 9^3+ ··· evaluates to the number Σi 1∞ 9− 9^í
According to me this is a quite peculiar way to use the verb 'to evaluate' (in the intro of the present text: "A series is thus evaluated by examining ...."); you can show sources? I only saw it, meaning: given an expression (denoting a number), find the decimal representation of its value, exact or approximated. -- Hesselp ( talk) 21:37, 2 May 2017 (UTC)--
There is a situation with Hesselp ( talk · contribs · deleted contribs · logs · filter log · block user · block log) on the page Series (mathematics) and the talk page Talk:Series (mathematics). He has been edit-warring to include his rewrite of the article [3], [4], [5], [6], [7], [8]. Although not at the moment above 3RR, the above is clear indication of edit warring, being reverted by four different editors. He was warned against edit warring, yet persists. Other editors have attempted to engage him at Talk:Series (mathematics), but attempts to resolve the dispute amicably are met with walls of antagonistic rambling text: [9], [10], [11], [12], [13], among others. We have given up on trying to interact with this user, in the spirit of WP:DENY (the above posts strongly suggest trolling). But I believe the time has come for this disruption to be put to an end administratively. (Pinging other involved editors: @ Hesselp:, @ D.Lazard:, @ MrOllie:, @ Wcherowi:.) Sławomir Biały ( talk) 11:58, 6 May 2017 (UTC)
Toegevoegd vraag aan L3X1 (?)
D.Lazard writes (15:14, 30 April 2017): "Presently, mathematicians agree on the concept of a series, but as usual for concepts that have many applications, the formal rigorous definition is too technical for being understood by beginners, .....".
This 'agree on' seems to be not in accordance with the ongoing rewriting of the Definition section in the article. Not with the absence of a decisive unambiguous source. And not with the result of a survey, made around 2008. About eighty books on calculus were inspected, the results are shown below (press [show]). The original language was not always English; capital-sigma forms were seen as not different from a1 + a2 + a3 + ··· .
Bowman, Britton/Kriegh/Rutland, Edwards/Penny, Open University-UK, Small/Hosack
2. An (infinite) series is an expression that can be written in the form a1 + a2 + a3 + ···
Anton/Herr, Anton, Anton/Bivens/Davis
3. An (infinite) series is a formal sum of infinitely many terms.
R A Adams
4. An (infinite) series is a formal infinite sum.
Ahlfors
5. The formal expression a1 + a2 + a3 + ··· is called an (infinite) series.
Matthews/Howell, Sherwood/Taylor
6. An (infinite) series is an indicated sum of the form a1 + a2 + a3 + ···
Kaplan
7. An (infinite) series is a sequence a1, a1 + a2, a1 + a2 + a3, ···
Hurley
8. An (infinite) series is a sequence whose terms are to be added up.
Marsden/Weinstein
9. An (infinite) series is the indicated sum of the terms of a sequence.
Daintith/Nelson, Kells, Weber
10. An (infinite) series is the sum of the terms of a sequence.
Wikipedia-Spanish
11. An (infinite) series is the sum of a sequence of terms.
Borowski/Borwein
12. An (infinite) series is the sum of an infinite number of terms.
Lyusternik/Yanpol'skii
13. An (infinite) series is a sum of a countable number of terms.
Borden
14. An (infinite) series is an infinite addition of numbers.
Goldstein/D C Lay/Schneider(/Asmar)
15. An (infinite) series is an infinite sum: a mathematical proces which calls for an infinite number of additions.
Davis/Hersh
16. An (infinite) series is a sequence of numbers with plus signs between these numbers.
Bers
17. We have an (infinite) series if, between each two terms of an infinite sequence, we insert a plus sign.
Maak
18. An (infinite) series is an ordered pair {an}; {sn} with sn short for a1 + a2 + … + an
Buck, Gaughan, Maurin, Protter/Morrey, Zamansky, Encyclopaedia of Mathematics1992,
Wikipedia-Dutch, Wikipedia-English, Wikipedia-French
Buck writes(1956,1965, 1978): "An infinite series is often defined to be 'an expression of the form Σ1∞ an '. It is recognised that this has many defects."
19. If we try to add the terms of an infinite sequence a we get an expression of the form a1 + a2 + a3 + ··· which is called an (infinite) series.
Stewart
20. If we add all the terms of an infinite sequence, we get an (infinite) series.
De Gee
21. When the terms of a sequence are added, we obtain an (infinite) series.
Croft/Davison
22. When we wish to find the sum of an infinite sequence <an> we call it an (infinite) series and write it in the form
a1 + a2 + a3 + ···
Keisler
23. Given a sequence a , then the sequence a1, a1 + a2, a1 + a2 + a3, ··· is called an (infinite) series.
Apostol, Burrill/Knudsen, Endl/Luh, Fischer, Forster, S R Lay, Rosenlicht, Wikipedia-Italian
24. Given a sequence a, then the sequence a1, a1 + a2, a1 + a2 + a3, ··· is called the (infinite) series
connected with the sequence a.
Barner/Flohr, Friedemann,
Dijkstra cs (Twente University), Almering (Delft University)
25. Given a sequence a, then the infinite sum a1 + a2 + a3 + ··· is called an (infinite) series.
Grossman, Leithold
26. Given a sequence a, then the expression a1 + a2 + a3 + ··· is called an (infinite) series.
L J Adams/White, Blatter, Van der Blij/Van Thiel, Gottwald/Kästner/Rudolph, Sze-Tsen Hu
27. Given a sequence a, the symbolic expression a1 + a2 + a3 + ··· we call an (infinite) series.
Rudin, Walter
28. Given a sequence a, an expression of the form a1 + a2 + a3 + ··· is an (infinite) series.
Thomas/Finney
29. No explicite attempt is made to describe the meaning of (infinite) series, although this term is used frequently.
Ackermans/Van Lint, Binmore, Cheney, Godement, Hille, Hirschman, Johnson/Kiokemeister, Knapp, Kreyszig, Larson/Hostetler, Lax, Morrill, Neill/Shuard, Riley/Hobson/Bence, Van Rootselaar, Ross, Varberg/Purcell/Rigden, Widder, Wikipedia-German, Duistermaat (Utrecht University), D&I (Groningen University)
30. For any sequence , the associated (infinite) series is defined as the formal sum (expression describing a sum) aM + aM+1 + aM+2 + ··· .
Wikipedia-Dutch (fall 2015)
31. An infinite sequence of real numbers is called (infinite) series. Original wording: On appelle 'série' une suite indéfinie de quantités (quantité: nombre reel).
C.-A. Cauchy.
This not very satisfactory situation, caused by the double meaning of 'convergence' in the 19th century, can be structured by accepting that:
- when 'series' is used denoting a mathematical object, it is synonym with 'sequence' (as in the 19th century and later), and
- in other cases 'series' is designating a certain kind/type of expression (or representation, or evaluation, or maybe even more).
Instead of 'series expression' mostly the shorter 'series' is used. But one has to realize that with 'convergent series' is not meant: 'the convergent mathematical object named series ', but: the convergent mathematical object denoted by the (type series) expression.
--
@Sławomir Biały. Please, present one or more explicit examples of occurrences of "antagonistic text" in my posts on this Talk page. And one or more examples of occurrences of "rambling text" in my posts on Talk page.
I hope I can learn from your examples, how to improve the presentation of my arguments. And how to avoid unnecessary blocking. --
To the list of "Secondary sources supporting Hesselp's edits" (22:52, 27 April 2017, answering Wcherowi's remark 17:16, 25 April 2017 "...your edits are not supported by citations to reliable secondary sources...") I add:
-
R. Creighton Buck (1920-1998, University of Wisconsin), Advanced Calculus, 1st ed. 1956, 2nd ed. 1965, 3rd ed. 1978
"An infinite series is often defined to be 'an expression of the form Σ1∞ an '. It is recognised that this has many defects."
--
Spivak: editions 1967, 1980, 1994, 20??(nog controleren)
"........an acceptable definition of the sum of a sequence should contain, as an essential component, terminology which distinguishes sequences for which sums can be defined from less fortunate sequences."
D.Lazard, 15:14, 30 April 2017(UTC)
...a series is a
mathematical object. It appears that this concept is not a simple one, as it involves the concept of
infinity, which was not well understood nor well accepted before the end of the 19th century (this make your citation of Cauchy irrelevant for discussing the modern terminology; note that he avoided carefully to talk about infinity).
Wikipedia "Mathematical object": A mathematical object is an abstract object arising in mathematics. .... In mathematical practice, an object is anything that has been (or could be) formally defined, ...
Victor J. Katz, A History of Mathematics An Introduction (reprint November 1998), p.705
It was Augustin-Louis Cauchy, the most prolific mathematician of the nineteenth century, who first established the calculus on the basis of the limit concept so familiar today. Although the notion of limits has been discussed much earlier, even by Newton, Cauchy was the first to translate the somewahat vague notion of a function approaching a particular value into arithmetic terms by means of which one could actually prove the existence of limits. Cauchy used his notion of limit in defining continuity (in the modern sense) and convergence of sequences, both for numbers and of functions. .......
I have no desire to enter long discussions about this article, but I wanted to leave a few comments about this revision [14]:
— Carl ( CBM · talk) 15:44, 8 May 2017 (UTC)
@Carl. Thank you very much for your concrete comments.
On point 1: I understand your remark. But......in this case? You add: "whenever possible". Here we have a mathematical object: (in modern words) a mapping on N. The traditional word for what later on is normally named "sequence". And we have a mathematical concept(?), a certain type of expression (a sign for the 'infinite summation function' plus a sign for a sequence as its argument). You may change the order of the two. The same 'series-type' we meet when classifying representations (for numbers or functions), and when classifying expansions (for functions).
I'm afraid this cannot be combined in one phrase. I explained this in my article text.
On point 2: I plan to smooth the content of this footnote. Maybe omit it completely. You are right that this sharp, maybe exaggerated wording is better suited for a discussion on Talk page.
On point 3: On the unusual spacing in R e d u c t i o n o f . . . you're 100% right, I was lazy when I copied it from elsewhere. On the use of other extra spacings: you cannot see them as making the text, and the formulas, better readable? Enough to accept some deviation from standard style?
And on the use of more 'ordinary prose': maybe a question of taste as well. I shall reconsider this. I wouldn't take as an example the present text of the article. For me that's very far from any encyclopedic style. --
The facts, in short: David Eppstein was 'baffled' (Talk page 22:01, 8 May 2017) by my incomprehension regarding the true nature of "expressions" and "infinite expressions" (being the central key-term in the definition of 'series'). After asking for the difference between finite and infinite expressions (09:38, 9 May, again 08:44, 10 May), the answer (14:36 and 15:43) was unclear to me, so I made my question more concrete (points A-E, 18:49, 10 May). Reaction by David Eppstein: "...no more interaction with you", "I see your edits as tendentious and disruptive" and some more not very positive remarks. --
D.Lazard's post in WP:ANI is copied here, in three parts with comments by Hesselp indented.
Hesselp's version of
series (mathematics) begins by "In mathematics (calculus), the word series is primarily used as adjective ...
". This is not only
WP:OR but also blatantly wrong: It suffices to look at any modern textbook of calculus to know that "series" is primarily used in mathematics as a noun.
Note also that, although series are studied in most textbooks of calculus the only source for Hesselp's lead is about 150 years old (and also misunderstood).
The remainder of Hesselp's version of the article continues in the same style and consists only of Hesselp's own inventions, beliefs and/or misinterpretation of the rare source that he produces. D.Lazard ( talk) 23:35, 10 May 2017 (UTC
Now we observe another attack toward " Series (mathematics)" (see "Relevant discussion at WP:ANI" above); User:Hesselp insists on a single definition of a series as a sequence (of terms). For now the article defines a series as (a special case of) an infinite expression. Another equivalent definition in use is, a pair of sequences (terms, and partial sums). Regretfully, this case is not covered by my "bastion", since the set of series is itself not quite an instance of a well-known mathematical structure (though some useful structures on this set are mentioned in our article). And still, it would be useful to write something like A person acquainted with series knows basic relations between terms and partial sums, and does not need to know that some of these notions are "primary", stipulated in the definition of a series, while others are "secondary", characterized in terms of "primary" notions. Implementation need not be unique. When several implementations are in use, should we choose one? or mention them all "with due weight"? or what? Any opinion? Boris Tsirelson ( talk) 16:40, 12 May 2017 (UTC)
The same is the case with the 'definition' in the present version of the article; implying the self-referring sentence: A series is an expression of the form ..+..+..+ ··· denoting a series.This seems a quote, but the word "denoting" does not appear in the article. This method of changing the wording of the content that he pretends discussing is systematic. This strongly suggests a bad faith; in any case it is definitively impossible to have a constructive discussion with this editor. Therefore, a permanent ban seems the only acceptable solution. D.Lazard ( talk) 21:18, 12 May 2017 (UTC)
The question whether or not a sound criterion exists to decide between finite expression and infinite expression is mentioned in the following posts:
21:50 2 May 2017,
09:38, 9 May 2017,
15:43, 10 May 2017,
18:49, 10 May 2017,
20:45, 10 May 2017,
22:19, 10 May2017.
No clear answer on this question is formulated yet.
(moet nog afgemaakt)
In mathematics (calculus), the word series is primarily". This is not only WP:OR but also blatantly wrong:"
used as adjective ...
The same is the case with the 'definition' in the present version of the article; implying the self-referring sentence: A series is an expression of the form ..+..+..+ ··· denoting a series.This seems a quote, but the word "denoting" does not appear in the article.
Comments on D.Lazard's post 23:35, 10 May 2017 :
On "..any modern textbook.." : For a survey of attempts to define 'series', see the list '32 attempts' in
this post. The 32 different wordings can be combined to a handful of really different content. Most of the about 80 authors say that a series IS an expression, but leave it to the reader to find out what's the character of the mathematical object, denoted (described, referred to) by this expression. The same is the case with the 'definition' in the present version of the article; implying the self-referring sentence: A series is an expression of the form ..+..+..+ ··· denoting a series.
And left to the reader as well is the question of how to interprete the label "convergent series". A convergent expression seems to be nonsense, but without any idea about the content of the expression, it's not easy to understand what's really denoted by this label.
In some sources (
Spivak,
Buck,
Dijksterhuis,
Van Rooij,
Cauchy,
Gauss) can be found more explicitely how to interprete the usual wordings. Making it possible to see the connection between the traditional - self-referring - wordings in most books on calculus, and the way how the label 'series' is used by mathematicians in practice.
Only a minimal change in interpretation is needed. That is: don't say: 'series' IS the expression ..+..+..+ ··· itself, but say: 'series' is used to label a certain TYPE OF expression. The type, constituted by a summation symbol (the sigma-sign, or the repeated pluses and end-dots) combined with the name of a sequence.
This is what should be an improvement of the article, with its consequences in the wording of the remaining standard content. Helping the reader to grab the meaning of the on-first-site strange combination 'convergent series' (= convergent expression).
Original Research ?: The explanation of the meaning of 'convergent series' - as being nothing else as summable sequence - is the very first statement in chapter 'Series' in Michael Spivak's well known "Calculus". Already for half a century: 1st ed. 1967, 4th ed. 2008. See More precise terminology 21:37, 9 May 2017
"The only source...."? No, all 80 rather modern calculus books in the list in this post, 20:28, 8 May 2017 served as sources. And of the 19th century sources are mentioned earlier: Cauchy, Susler, Itzigsohn, Gauss, Von Mangoldt. Why doesn’t D.Lazard mentions which one of this five he has studied, and which point in it I should have misunderstood?
The remainder of Hesselp's version.... Without concrete examples, I can't comment on D.Lazard's last sentence. Is it the conclusion of everyone who have read this edit? --
Attempting to find a way to some kind of consensus, I add the following lines to this Talk page.
Citations, taken out of longer posts on
Wikipedia talk:WikiProject Mathematics
-
Tsirel - 19:15, 12 May 2017: ".. in general an expression has no value (but in "good" cases it has);" (Comment Hesselp: the dispute is about the question whether a series-type expression has (in "good" cases) a number as its value, or a series (For: "a series is denoted by an expression like ..+..+..+···"))
-
CBM - 20:00, 12 May 2017: "... the definitions that are often given in the books lack something that would be present in a graduate level text." (Comment Hesselp: No one has presented such a graduate level text in this Talk page.)
-
CBM - 20:00, 12 May 2017: "...we should follow the sources and present the same general understanding that they convey.] (Comment Hesselp: That's easier said than done, see survey in
09:38, 9 May 2017)
-
CBM - 20:09, 12 May 2017: "If numerous sources all find it possible to discuss a concept without a formal definition, we can certainly do so as well."
-
D.Lazard - 20:43, 12 May 2017: " In any case, a series is not a sequence nor a pair of sequences nor an expression. It is an object which is built from a sequence." (Comment Hesselp: D.Lazard's edited since
09:50, 14 Februari 2017 seven times a version with: "a series is an expression").
-
Tsirel - 05:02, 13 May 2017: "What does it mean? A vague term whose meaning is determined implicitly by the context, case-by-case?"
-
Taku - 23:10, 13 May 2017: "... a series is a more of a heuristic concept than an explicitly defined concept."
Observations Studying the terminology used in the 19th (and a good part of the 20th) century, concerning the 'series-representation' of numbers (and of functions), we can see two noteworthy points.
(1) The word 'series' was used frequently in situations where we should use 'sequence' now. (Also German 'Reihe' in 'Folge'-situations, and French 'série' in 'suite'-situations.) Cauchy introduces 'série' explicitely for a sequence with numbers as terms; much later Bourbaki seems to copy this by using 'series' for a sequence with terms allowing the existence of a 'sum series'. The names 'arithmetical series', 'harmonical series', 'Fibonacci series', etc. were in common use. (2) The words converge/convergent/convergence were used in case the terms have a limit, as well as in case the partial sums have a limit. Cauchy seems to use the verb 'converger' for terms with a limit, and the adverb 'convergent' for partial sums with a limit; quite confusing. And Gauss once remarks: (Werke Abt.I, Band X, S.400) "Die Convergenz einer Reihe an sich ist also wohl zu unterscheiden von der Convergenz ihrer Summirung ...." (The convergence of the sequence itself has to be distinguished from the convergence of its summation.)
Suppostion This situation: two words (series and sequence) for one notion, and one word for two properties (limiting terms and limiting partial sums), caused ongoing confusion. More and more culminating in a belief in the existence of a third 'mathematical object', apart from 'sequence' and 'the sum sequence of a given sequence'. A mysterious object or notion, whose definition/description causes the difficulties mentioned in the citations above.
How about the idea of describing this historical roots of the present confusion, in the Wikipedia article? Can this be seen as a description of the existing situation, or is this seen as OR? --
I don't think the article should focus on the historical roots to any great extent, except perhaps in a section on history. Sources from the 19th century are not likely to be of much use in this kind of elementary article, and indeed there were many more terminological problems at that time (compare the common use of "infinitesimal" at that time). Every contemporary calculus book I have seen has the same concept of a series, although of course the wording may vary from one author to another. — Carl (CBM · talk) 01:21, 15 May 2017 (UTC)
- Instead of the heading "Definition", I have in mind: "Names and notations".
- About recent changes in the text of the article:
• The self-referring "A series is an expression denoting a series" can't be found in the text any longer. Improvement.
• In the definition of 'series', the two-track construction "a series is an infinite sum, is an infinite expression of the form .." disappeared. Improvement.
• The "such as" regarding the capital-sigma notation. Improvement. (Maybe some more variants can be shown? As well as
a1 + a2 + ... + an + ... as variant of the pluses-bullets form.)
• The label "infinite expression" (instead of "expression") is still there. Although no criterion is found for decerning. See
, , .
• The intro (almost at the end) says: "When this limit exists, one says that the series is convergent or summable, and the limit is called the sum of the series. And the present definition says: "a series is an infinite sum,..". Combined we get wordings as: "a summable infinite sum" and "the sum of an infinite sum".
I know there are books where you can find this; but it's not very nice and comprehensible. Is it definitely OR to add that it's not unusual to say "summable sequence" and "sum of a sequence" as well? I referred to Spivak (1956...2008) and many hits in Google.
The third sentence in the present text says: "Series are used in most areas of mathematics,..". Isn't it true that the content of this sentence can be worded as well by: "Capital-sigma expressions and pluses-bullets expressions are used in most areas of mathematics".
Why are this notations so important? Because they express a method to denote/describe irrational numbers (and as an generalization also functions) by means of a regular-patterned sequence with more familiar rationals as terms (or 'easier' functions).
The usual word for such a method to describe mathematical objects by means of simpler objects, is "representation". We have: the decimal representation, the continued fraction representation, the infinite product representation, and some more. Not the least important is, what could be called "the infinite sum representation" or - in honour of the famous term - "the series representation". The representation based on the summation function for infinite sequences.
So, instead of saying "series are important" (with the hard to define term 'series'), you could say "the series representation is important" (describable without mysterious words). Is this a so big change that you are going to react with: "impossible, clear OR" ?
Last remark. Caused by personal circumstances I've to tell that I leave by now Wikipedia for at least a couple of weeks. I wish you fruitful discussions. Hessel Pot --
1. The original wording by Saidak can be of interest on a special Saidak-page, not on the page titled "Euclid's theorem".
2. I cannot see why the 7-lines version of the proof should be more accessible for laymen than the short version. For:
2a. The central argument in the proof is described in essentially the same way in both versions: "For any n > 1, n and n + 1 have no common factors; they are
coprime" versus "Because all prime divisors of a natural number n are different from the prime divisors of n+1, . . . ".
2b. The short version doesn't have the "induction for any k".
2c. Nor the superfluous and difficult to read and interprete "example" at the end.
2d. Using N2, N3, ..., but not N1, is confusing.
2e. Why 'at least' in '1806 has at least four different prime factors' ?
Who refutes this arguments? Who has a better proposal for the text of the proof? --
Hesselp (
talk) 21:42, 5 April 2017 (UTC)
Since each natural number (≥2) has at least one prime factor, and two successive numbers n and (n+1) don't have any prime factor in common, the product n×(n+1) has more different prime factors than the number n itself.
This implies that each term in the infinite sequence: 1, 2 (1×2), 6 (2×3), 42 (6×7), 1806 (42×43), (1806×1807), (1806×1807) × (1806×1807 + 1), · · · has more different prime factors than the preceding.
The sequence never ends, so the number of different primes never ceases to increase. --
===Second attempt plus === Improved wording of Saidak's proof; discussion in Talk.
Since each natural number (≥2) has at least one prime factor, and two successive numbers n and (n+1) don't have any factor in common, the product n×(n+1) has more different prime factors than the number n itself. So in the chain of
pronic numbers:
1×2 = 2 {2}, 2×3 = 6 {2, 3}, 6×7 = 42 {2,3, 7}, 42×43 = 1806 {2,3,7, 43}, 1806×1807 = 3263443 {2,3,7,43, 13,139}, · · ·
the number of different primes per term will increase forever. --
I'm pleased to see that Lazard (Febr.14, 2017, line 4) describes the meaning of the word series as an expression of a certain type. Less clear (or better: mysterious) is the remark: "obtained by adding together all terms of the associated sequence"; what could be meant by "adding together"? What kind of action should be performed, by who, on which occasion, to obtain / create an expression of the intended kind?
More remarks on the present text of the article, in this Talk page: 15:14 16 April 2017.
To get things clear, I propose to start this article in about the following way:
I n t r o d u c t i o n
In mathematics (
calculus), the word series is primarily used for
expressions of a certain kind, denoting
numbers (or functions).
Symbolic forms like and or expressing a number as the limit of the partial sums of sequence , are called series expression or shorter series.
Secondly, in a more abstract sense, series is used for a certain kind of representation (of a number or a function), and also for a special type of such a series representation named series expansion (of a function, e.g. Maclaurin series, Fourier series).
And thirdly, series can be synonymous with sequence.
Cauchy defined the word series by "an infinite sequence of real numbers".[source: Cours d'Analyse, p.123, p.2, 1821, 2009]
This use of the word 'series' can be seen as somewhat outdated.
The study of series is a major part of
mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures (such as in
combinatorics), through
generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as
physics,
computer science,
statistics and
finance.
C o n t e n t s
D e f i n i t i o n s, c o m m o n w o r d i n g s
Given a infinite sequence with terms et cetera (or starting with ) for which addition is defined, the sequence
is called the sequence of partial sums of sequence .
Alternative notation: .
Example: The sequence 1, 2, 3, 4, ··· is the sequence of partial sums of sequence 1, 1, 1, 1, ··· ; the sequence 1, 1, 1, 1,··· is the sequence of partial sums of sequence 1, 0, 0, 0,··· ; this can be extended in both directions.
A series is a written expression using mathematical signs, consisting of
- an expression denoting the function that maps a given sequence on the limit of its sequence of partial sums
combined with
- an expression denoting an infinite sequence (with addition and distance defined).
Second meaning The symbolic forms (plusses-bullets form) and (capital-sigma form)
are sometimes used to denote the sequence of partial sums of sequence , instead of the value of its eventual existing limit.
A sequence is called summable iff its sequence of partial sums converges (has a finite limit, named: sum of the sequence).
Convergent / divergent series The combination convergent series shouldn't be interpreted literally, for an expression itself cannot be convergent or divergent. By tradition "Σ is a convergent series" as well as "series Σ converges" are used to express that sequence is summable. Similarly, "Σ is a divergent series" and "series Σ diverges" are used to say that sequence is not summable.
Convergence test for series Again, this traditional wording cannot be taken literally because 'series' is the name of an expression of a certain kind, not the name of a mathematical notion. An alternative is: summability test for sequences.
Absolute convergent series This is the traditional naming for a sequence with summable absolute values of its terms. The alternative absolute summable sequence is not in common use.
Series Σ and sequence are interchangeable in traditional clauses like:
- the sum of series Σ , the terms of series Σ , the (sequence of) partial sums of series Σ ,
the
Cauchy product of series Σ and series Σ
- the series Σ is geometric, arithmetic, harmonic, alternating, non negative, increasing (and more).
There is no standard interpretation for the limit of series Σ .
S e r i e s r e p r e s e n t a t i o n o f n u m b e r s a n d f u n c t i o n s
In some contexts the word 'series' shouldn't be seen as referring to a certain type of written symbolic expressions, but as referring to a special type of representation of numbers (and functions). Namely: defining a (
irrational) number as the limit of the partial sums of a known infinite sequence of (rational or irrational) numbers. And in the case of functions: defining a function as the limit of the partial sums of an infinite sequence of functions (which are seen as 'easier' or more elementary in one way or another than the function represented by the limit).
Examples of the use of the word 'series' in this sense, can be seen in the final sentences of the introduction above, starting with "The study of series is a major part ...".
As comparable with the idea of series representation or infinite sum representation can be seen: the continued fraction representation and the infinite product representation (for numbers and functions).
S e r i e s e x p a n s i o n o f f u n c t i o n s
The combination 'series expansion' is used for a special type of series representation of functions. ('Series expansion of numbers ' is meaningless.)
A series expansion is a representation of a function by means of the infinite sum of a sequence of power functions of increasing degree, in one of its variables. Or functions like (for example) .
The labels Maclaurin series, Taylor series, Fourier series shouldn't be seen as denoting expressions but rather representations of the type series expansion. So Maclaurin series should be understood as Maclaurin expansion, Fourier series as Fourier expansion, et cetera. [Source: WolframMathWorld
series expansion and
Maclaurin series].
P o w e r s e r i e s
"Power series" can be used
- as synonym for "Maclaurin expansion", and
- for a series expression which includes a sequence of power functions with increasing degree.
C a u c h y a s s o u r c e o f c o n f u s i o n
Cauchy, in his 'Cours d'Analyse' (1821) made an important, but quite subtile, distinction between the meaning of 'to converge' and 'being convergent':
- a sequence (French: suite) can converge (both French and English) to a limit, versus
- an infinite sequence of real numbers (named 'série' by Cauchy) having its sequence of partial sums converging to a limit, the first sequence named 'une série convergente ' .
Only a tiny difference between 'sequence' and 'series', but an essential one between 'converging' and 'convergent'.
This imprudent choise caused permanent confusion around the use of the word 'series'(e.g. in the German translations of 'Cours d'Analyse' of 1828 and 1885), until the present day.
[sources:
Cauchy, see p.123 and p.2 quantité
C.L.B. Susler, 1828,
Susler, S.92,
Carl Itzigsohn, 1885,
Bradley/Sandifer, 2009 ]
[More sources on the problem with 'series' in books/publications by: professor H. Von Mangoldt, E.J. Dijksterhuis, H.B.A. Bockwinkel, professor N.G. de Bruijn, professor A.C.M. van Rooij, professor D.A. Quadling, Mike Spivack, H.N. Pot; links have to be added. Several of this sources are written in Dutch.]
Overwegen om delen van de rest van dit artikel onder te brengen op andere pagina's. Vaak staat er nu al een verwijzing naar "Main article".
Rekenregels Rewriting voor serie expressions
To 166.216.158.233, and ... . On Februari 28 2017, you changed {sk} into (sk) at several places. I understand your argument (a sequence is a mapping, not a set), but I see your solution as insufficient. For without any harm, you can do without braces/parentheses at all, and without any index symbol as well. A sequence is defined as a mapping on the set of naturals, so label them with a single letter. Just as people mostly do with mappings/functions with other sets as domain: f, g, F, G, ... .
When there is a risk of confusion you can write "sequence s", "sequence S" in stead of just "s" or "S".
Who has objections? (Yes, I know the index is tradition, but it is superfluous and therefore disturbing.)
In the Definition section, three lines after "More generally ..." I read:
the function is a sequence denoted by .
I count three different notations for the same domain--function (sequence), four lines later a fourth version - - is used.
Last remark: It's not correct to say that sequences ( and ) are subsets of semigroup . --
For me it is impossible to find any information in the second part of subsection 'Definition'- after 'More generally....'.
The text seems to suggest that the notion of "series" (whatever that is ...) can be extended from something associated with sequences (mappings on the set of naturals) to a comparable 'something' associated with mappings on more general index sets. But nothing is said about how such generalized mappings can be transformed into a limit number . Is it possible to generalize the tric with the 'partial sums'? This index sets has to be countable? No reference is given. (The present text is composed by Chetrasho
July 27, 2011).
I propose to skip the text from 'More generally' until 'Convergent series'. Any objections? --
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My editing priorities: 1.Correct and non-misleading statements 2.Sufficient referencing 3.Accessible leads 4.Article structure
Hello Bill Cherowitzo. You are right, my two questions are answered in the final section of the article.
But I persist that the description of the notion named series becomes even more unclear by adding six sentences (the greater part of the Definition section) on a generalization that will be unknown to most readers.
Moreover, the correlation between the position of this notion connected with sequences, and its position connected with mappings on an index set, is not very strong. For:
In (elementary) calculus two different symbolic forms (both named 'series') are used, expressing the relation between a sequence and its 'sum'. One of them, the plusses-bullets form cannot be used in the generalized situation. And the other one, the capital-sigma form needs adaption ( instead of or or or ).
The absence of relevant information in this six sentences is not undone by a 'lack of information tag'. Skipping this sentences I cannot see as a "removal of [relevant] material". --
Openingszin datum ???
Infinite series
The sum of an infinite series a0+a1+a2+... is the limit of the sequence of partial sums
S N = a 0 + a 1 + a 2 + . . . + a N {\displaystyle S_{N}=a_{0}+a_{1}+a_{2}+...+a_{N}} {\displaystyle S_{N}=a_{0}+a_{1}+a_{2}+...+a_{N}} ,
as N → ∞. This limit can have a finite value; if it does, the series is said to converge; if it does not, it is said to diverge. The fact that infinite series can converge resolves several of Zeno's paradoxes.
Openingszin 30 March 2006 16:48
In
mathematics, a series is often represented as the
sum of a
sequence of
terms. That is, a series is represented as a list of numbers with
addition operations between them, e.g,
which may or may not be meaningful, as it will be explained below.
Yesterday's (13 April 2017) reduction in this section is an improvement, yes. Now this shorter version makes it easier to explain my objection to its central message. I paraphrase this message in the next four lines:
1. For any sequence is defined a
2. associated series Σ (defined as: an ordered "element of the free abelian group with a given set as basis" - the link says).
3. To series Σ is associated the
4. sequence of the partial sums of .
Why in line 2 an 3 a detour via a double 'association'(?) with something named 'series'? Is the meaning of that word clearly explained in this way to a reader? I don't think so. I'm working on a text that starts with:
"In mathematics the word series is primarily used for expressions of a certain kind, denoting numbers (or functions). Secondly"
I plan to post this within a few days. --
Hesselp (
talk) 13:39, 14 April 2017 (UTC).
Who can tell me how to find out whether or not a given "ordered element of the free abelian group with a given set as basis" has 100 as its sum? Who can mention a 'reliable source' where the answer can be found?
Why should this mysterious serieses be introduced at all, in a situation where it's completely clear what it means that a given sequence has 100 as its sum. I cannot find a motivation for this in a 'reliable source' mentioned in the present article.
So skip this humbug (excusez le mot).
About an eventual 'immediate removal': Should I have to expect that a majority in the Wiki community will support removing a serious attempt to describe in which way (ways!) the word 'series' is used in most existing mathematical texts. And replace a version including a 'definition' which has nothing to do with the way this word is used in practice; only because the wording has some resemblance with meaningless wordings that can be found in (yes, quite a lot of) textbooks.
In the present 'definition' of series the words 'formal sum' are linked to a text on Free abelian groups. Can this be seen as a 'reliable source' for a reader who wants to know what could be meant by 'formal sum'? Wikipedia is not open for attempts to improve this? --
Line 3, quotation: "Summation notation....to denote a series, ..."
A notation to denote an expression ?? Sounds strange (first sentence says: series = expression of certain kind).
Line 4, quotation: "Series are formal sums, meaning... by plus signs),"
I can read this as: "The word 'sum' has different meanings, but the combination 'formal sum' is a substitute for 'series' (being forms consisting of sequence elements/terms separated by plus signs)". Correct?, this is what is meant?
But "Series are formal sums" seems to communicate not exactly the same as " 'series' is synonym with 'formal sum' ".
Line 4-bis, quotation: "these objects are defined in terms of their form"
With 'these objects' will be meant: 'these expressions (as shown in the first sentence)', I suppose. But then I miss the sense of this clause. An expression IS a form, and don't has to be defined (or described?) in terms OF its form.
Line 6. Properties of expressions? and operations defined on expressions? This regards operations as enlarging, or changing into bold face, or ...?
Line 7. "...convergence of a series". In other words: "convergence of a certain expression"? I'm lost.
I'll show an alternative. --
=== Three proposals for adaptations in the Definition section ===
I. Note 3 in the present text, saying "...a more abstract definition....is given in....", should be removed.
For it doesn't have any sense to refer to a 'more abstract definition' of
an expression of the form , labeled with the name 'series'.
There is not a 'less abstract definition' of this kind of expression either. Only a description.
II. A more direct formulation of the third sentence in this section is:
"A series is also called formal sum, for a series expression has a well-defined form with plus signs."
III. Remarks on the 'usefullness' and the 'fundamental propery' of such expressions of the form , shouldn't be included in a definition section. --
H o w t o r e d u c e c o n f u s i o n
The best thing to do is: Stop using the word 'series' at all, and say:
(absolute) summable sequence and summability tests, in stead of: (absolute) convergent series and convergence tests .
Second best is: inform students and readers of Wikipedia about the historical source of the confusion. Let them understand that the existence of any definable notion 'series' (different from 'sequence') is a wide-spread misconception. And train them to interprete (absolute) convergent series as nothing else as summable sequence. --
Answer to Lazard: Thank you for contributing to the search for the best way to describe what is meant with the word 'series' in texts on mathematics(calculus). I saw some points in your rewriting of the Definition section which I can see as improvements. But there are some problems left:
1. Rewording the first sentence more close to the usual way as definition of 'infinite series / series', I get:
An infinite sum is called series or infinite series if represented by an expression of the form: where ...
This paraphrasing is correct?
Please add an explanation of what you mean by 'infinite sum'. &nbnp;And tell how a blind person can decide whether or not he is allowed to say 'series' to such an infinite sum, as he cannot see the form of the representation.
2. In the third sentence 'summation notation' is introduced, showing a 'capital-sigma' form, followed by an equal sign and a 'plusses-bullets' form. Why two different forms to illustrate the 'summation notation'?
3. Please explain what you mean with 'formal sum' (fourth sentence). See
this discussion. And the same question for 'summation' at the end of that sentence.
4. Your seventh sentence end with "...the convergence of a series". Do you really mean to define "the convergence of an expression(of a certain type)?
5. Finally, I'ld like to see an explanation of the clause "the expression obtained by adding all those
[an infinite number of] terms together" (fifth sentence in the intro). I don't see how the activity of 'adding' (of infinite many terms!) can have an 'expression' as result. --
A sum of two numbers given in series representation,
a product of two numbers given in series representation, and
a product of two numbers, one of them given in series representation,
can be reduced according to:
The same applies for functions instead of numbers. --
Invoeren van mijn alternatief, tot aan "Examples", te wijzigen is "Examples of the use of the series representation', of 'series expressions', and of the word 'series' in different situations" .
Adjective in combination with: 'expression', 'representation', expansion'.
Noun as - a contraction of 'series expression'
-as synonym with 'seqence',
- as part of traditional wordings: (absolute) convergent series, convergence test for series, Cauchy product of two series, meaning ............
Jeder Folge ist identisch mit der Partialsummenfolge seiner Differenzenfolge, also: 'Reihe' und 'Folge' sind synonym.
Deswegen sagt Satz 4 der Definition:
"Falls die Folge/Reihe konvergiert, so nennt man die Grenzwert der Folge/Reihe auch Summe der Folge/Reihe " .
Korrekt? Was ist hier definiert?
Zur Zeit wird eine Alternative diskutiert im 'Talk page' der englische
Wikipedia.
--
Hej/Hallo. Hier versucht ein Holländer auf Deutsch zu schreiben.
Ich (*1942) bin schon sehr lange interessiert in die Frage um das Unterschied (falls existierend) zwischen 'Folge' und 'Reihe'. Heute entdeckte ich diese Wiki-Seite, wo ich meine Frage sehr ausführlich behandelt sehe. Aber am Ende lese ich doch wieder im Definition (Reihe):
'Reihe' ist die Nahme der Schreibweise (in Symbolform) für (in Textform) "die Partialsummenfolge der Folge ".
Folglich lese ich im Definition (Grenzwert einer Reihe):
Die Grenzwert einer Schreibweise , ist der Limes ......
Meine ewige Frage bleibt: Wie kann eine Schreibweise (englisch: a symbolic expression) einen Grenzwert haben? (oder: konvergent sein, oder eine Summe haben, oder konvergieren, oder ...). Das ist doch Unsinn?
Für meine Versuche die Sache auf zu klären, sehe
Ganz klar ? oder
An attempt to clarify the 'series' mystery
--
Hesselp 12:05, 19. Apr. 2017 (CEST)
Die Definition von 'Reihe' im heutigen Text lautet:
Für eine reelle Folge ist die Reihe die Folge aller Partialsummen .
Hier wird (meiner Meinung) nichts anderes gesagt als:
Mit anderen Worten:
ist eine (kürzere) Schreibweise für die Partialsummenfolge einer reelle Folge ;
ein solcher Ausdruck - die Sigma-Schreibweise für Folgen - wird Reiheform oder Reihe genannt.
Korte titel: Paraphrasierung der Reihe-Definition.
Verdediging bij skippen: Es soll zumindestens erklärt werden warum die Paraphasierung nicht richtig ist.
Bij reactie "overbodig": oude regel vervángen door de nieuwe. De oude is impliciet en daarom erg onduidelijk.
Stephan Kulla Reaktion zu "sollte anstelle von einer Reihe, von einer Folge sprechen?
Tja, eigentlich: ja!. Oder....obwohl...historisch gesehen sind die Wörte 'Reihe' und 'Folge' sehr oft wie Synonyme gebraucht (Und auch heute noch: sehe z.B.
[1] "Folge und Reihe sind also nicht scharf voneinander trennbar. Die
Zeitreihen der Wirtschaftswissenschaftler sind eigentlich Folgen."
Und sehe
Esperanto: Rimarko: Ne ekzistas formala diferenco inter la nocioj
vico kaj serio. )
Darum kann man auch vorschlagen beide Wörte durcheinander zu benutzen.
Nochmals:
Die 'Folge der Partialsummen einer Folge' ist wieder eine Folge.
Der Limit einer 'Folge der Partialsummen einer Folge' ist ein Zahl.
E s g i b t k e i n m a t h e m a t i s c h e B e g r i f f d a z w i s c h e n / d a n e b e n .
Sehr viele Autoren schreiben und sprechen als ob es ein solcher Zwischenbegriff (ich sagte: 'Gespenst') gibt. Aber wenn man genau analysiert wie das 'Zwischenbegriff' (vielmals 'Reihe' genannt) definiert wird, dann kommt man nicht weiter als: "die Sigma-Schreibweise für Folgen". (Und dazu meistens auch: die "plusses/bullets notation" a1+a2+a3+··· . (Plusse/Punkte-Schreibweise ?))
Reaktion zu: "Ich verstehe das Problem noch nicht."
Das Wort 'Reihe' wird definiert als die Nahme einer symbolisch geschrieben mathematische Ausdruck (die Sigma-Schreibweise). Aber überall im Text wird gesprochen von 'konvergierende Reihe', 'konvergente Reihe', 'Summe der Reihe', 'Partialsummen der Reihe', usw. Wobei nirgends gesagt wird das der Wikibook-Leser die Schreibweise-Definition vergessen muss ('Reihe' wird niemals in dieser Auffassung gebraucht), und das traditionelle 'Summe der Reihe' muss lesen wie 'Summe der Folge'. Und 'konvergente Reihe' wie 'summierbare Folge'. Es kann dabei darauf hingewesen werden das Cauchy sehr unvorsichtig 'convergente' wählte als adjective für eine Zahlenfolge mit konvergierende Partialsummen (eine summierbare Zahlenfolge).
--
Part of a series of articles about |
Calculus |
---|
In mathematics (
calculus), the word series is primarily used as adjective specifying a certain kind of
expressions denoting
numbers (or functions).
Symbolic forms like and or expressing a number as the limit of the partial sums of sequence , are called series expression. 'Series expression' is often shortened to just 'series'.
Secondly, series is used as an adjective in series representation, denoting the kind of representation (of a number or a function) as a limit of the partial sums of a given sequence.
Thirdly, series is used, again as an adjective, in series expansion. Being a special type of series representation (of functions, not numbers). For instance:
the Maclaurin expansion of a given function and the Fourier expansion of a given function are series expansions.
Finally, (the noun) series can be synonymous with sequence. Cauchy defined the word series by "an infinite sequence of real numbers". [1] The use of the word 'series' for 'sequence' has a long tradition, with analogons in other languages, but seems to be considered as somewhat outdated.
The rather widespread idea about the existence of a mathematical notion (a definable mathematical object, called 'series'), 'associated' in some way with a given number sequence, with its partial sums sequence, and with the eventual limit thereof, is false.
The study of the series representation is a major part of
mathematical analysis. With this tool,
irrationals can be described/defined by means of (the limit of) a relatively easy descriptable sequence of rationals.
This kind of representation is used in most areas of mathematics, even for studying finite structures (such as in
combinatorics), through
generating functions. In addition to their ubiquity in mathematics, the series representation is also widely used in other quantitative disciplines such as
physics,
computer science,
statistics and
finance.
Given a infinite sequence with terms et cetera (or starting with ) for which addition is defined, the sequence
is called the sequence of partial sums of sequence .
Alternative notation: . Alternative name: the sum sequence of (sequence)
[2].
Example: The sequence (1, 2, 3, 4, ···) is the sum sequence of (1, 1, 1, 1, ··· ); being the sum sequence of (1, 0, 0, 0, ··· ); this can be extended in both directions.
A series, short for series expression, is a written expression using mathematical signs, consisting of
- an expression denoting the function that maps a given sequence on the limit of its sum sequence, combined with
- an expression denoting an infinite sequence (with addition and distance defined).
Examples: (plusses-bullets notation), (capital-sigma notation).
Sometimes, the same symbolic forms are used to denote the sum sequence of , instead of the value of its eventual limit.
A sequence with a converging sum sequence is called summable. The finite limit is called sum of the sequence.
A valid series expression has a summable sequence as its argument (and denotes a value). Otherwise the expression is void. Traditional wordings are: "convergent/divergent series expression" or "convergent/divergent series".
Convergent / divergent series The combination convergent series shouldn't be interpreted literally, for an expression itself cannot be convergent or divergent. By tradition "Σ is a convergent series" as well as "series Σ converges" are used to express that is summable. Similarly, "Σ is a divergent series" and "series Σ diverges" are used to say that is not summable.
Convergence test for series Again, this traditional wording cannot be taken literally because 'series' is the name of an expression of a certain kind, not the name of a mathematical notion. An alternative is: summability test for sequences.
Absolute convergent series This is the traditional naming for a sequence with summable absolute values of its terms. The alternative absolute summable sequence is not in common use.
Series Σ and sequence are interchangeable in traditional clauses like:
- the sum of series Σ , the terms of series Σ , the (sequence of) partial sums of series Σ , the
Cauchy product of series Σ and series Σ
- the series Σ is geometric, arithmetic, harmonic, alternating, non negative, increasing (and more).
There is no standard interpretation for the limit of series Σ .
In some contexts the word 'series' shouldn't be seen as referring to a certain type of written symbolic expressions, but as referring to a special type of representation of numbers (and functions). Namely: defining a (
irrational) number as the limit of the partial sums of a known infinite sequence of (rational or irrational) numbers. And in the case of functions: defining a function as the limit of the partial sums of an infinite sequence of functions (which are seen as 'easier' or more elementary in one way or another than the function represented by the limit).
As comparable with the idea of series representation (or: infinite sum representation) can be seen: the continued fraction representation and the infinite product representation (for numbers and functions).
R e d u c t i o n o f s u m s a n d p r o d u c t s
A sum of two numbers given in series representation,
a product of two numbers given in series representation, and
a product of two numbers, one of them given in series representation,
can be reduced according to:
(sequence or sequence summable)
.
The same applies for functions instead of numbers.
The name
'series expansion' is used for a special type of series representation of functions. (Not applicable to numbers.)
A series expansion is a series representation of a function, using a sequence of power functions of increasing degree, in one of its variables. Or functions like (for example) .
The labels Maclaurin series, Taylor series, Fourier series shouldn't be seen as denoting expressions but rather representations of the type series expansion. So Maclaurin series should be understood as Maclaurin expansion, Fourier series as Fourier expansion, et cetera.
[3]
The name
power series can occur
- as synonym for Maclaurin expansion, and
- denoting a series expression which includes an expression for a sequence of power functions with increasing degree.
Cauchy, in his 'Cours d'Analyse' (1821) made an important, but quite subtile, distinction between the meaning of 'to converge' and 'being convergent':
- A sequence (French: suite) can converge to a limit.
- A sequence with converging partial sums, is called convergent by Cauchy (meaning 'summable')
Moreover, an infinite sequence with real numbers as terms, he called a series (French: série).
This imprudent choise caused permanent confusion around the use of the word 'series' (e.g. in the German translations of 'Cours d'Analyse' of 1828 and 1885)
[4] until the present day.
Below, the words 'series' and 'convergent / divergent' are not always used conform the preceding descriptions. In such cases the context has to be taken into account to track down the intended meaning.
[More sources on the problem with 'series' in books/publications by: professor H. Von Mangoldt, E.J. Dijksterhuis, H.B.A. Bockwinkel, professor N.G. de Bruijn, professor A.C.M. van Rooij, professor D.A. Quadling, Mike Spivack, H.N. Pot; links have to be added. Several of this sources are written in Dutch.] --
Hesselp (
talk) 15:38, 16 April 2017 (UTC)
H o w t o r e d u c e c o n f u s i o n
The best thing to do is: Stop using the word 'series' at all, and say:
(absolute) summable sequence and summability tests, in stead of: (absolute) convergent series and convergence tests .
Second best is: inform students and readers of Wikipedia about the historical source of the confusion. Let them understand that the existence of any definable notion 'series' (different from 'sequence') is a wide-spread misconception. And train them to interprete (absolute) convergent series as nothing else as (absolute) summable sequence. --
Hesselp (
talk) 13:12, 17 April 2017 (UTC)
The present text strongly suggests that there is only one correct interpretation of what is meant by the word 'series' in mathematical texts. That is that the word 'series' is the name for a certain idea / notion / conception / entity. But what IS "it"?
"It" is NOT a number.
"It" is NOT a sequence (= a mapping on N)
"It" is NOT an expression (for the present text says: "a series is represented by an expression)
"It" is NOT a function.
"It" is 'associated' (what's that?) with a sequence. "It" is sometimes 'associated' with a value.
"It" has terms and partial sums.
"It" can have a limit, a value, a sum.
"It" can be geometric, arithmetic, harmonic, alterating, converging, diverging, absolute converging, and more.
What's in fact the content of this black "it"-box? It seems to be empty.
I'm going to replace this unsatisfactory text by an alternative introduction. Chiefly identical with what was shown in this Talk page
here, 18 April 2017. The only reaction on it was the remark that "Hesselp doesn't understand what A SERIES IS (in mathematics)". I agree with that. --
It seems like you have identified a Dutch school of thought on this topic. This would probably be good for a paragraph in the article, but certainly not a rewrite.--Bill Cherowitzo (talk) 05:20, 27 April 2017 (UTC)
About: expressing a number or a function by means of an infinite series. See:
[2]
The present text presents in the intro plus subsection Definition, four different 'definitions', all of them using the wording:
"a series IS ..." .
1. (Intro, sentence 1) "a series IS ... the sum of the terms of ..."
(Being the sum of numbers again a number, the words 'series' and 'number' are synonym.)
2. (Intro, sent.5) "The series of (associated with) a given sequence a IS the expression a1+a2+a3+··· "
(The word 'series' used as the name of a mapping.)
3. (Definition, sent.1) "a series IS an infinite sum, which is represented by a written symbolic expression of a certain type."
(It isn't clear whether or not the clause after the comma is part of the definition. 'IS' a series still an infinite sum, in situations where it is not represented by an expression of the named form?)
4. (Definition, sent.6) "series(pl) ARE elements of a total algebra of a ring over the monoid of natural numbers over the a commutative ring of the a's "
(The word 'series' as the name for elements of a certain structure; just as the word 'number' is used as the name for elements of another mathematical structure. To which element in this 'definition' is referred by the a's ? )
In case it is true, that the word 'series' has four different meanings in mathematics (is used in four different ways) the article headed by "Series" should be structured like:
a. The word 'series' is used as name/label for ......... .
b. The word 'series' is also used as name/label for ......... .
c. The word 'series' is used as name/label for .......... as well.
d. Moreover, sometimes the word 'series' is used as name/label for ......... .
The present text directs the reader to believe that there is ONE and only ONE sacred given-by-God-meaning of this word.
That's religion, not mathematics.
Do you think, Wcherowi, the summing up of different meanings is wrong?
Do you think, D.Lazard, the summing up of different meanings is wrong?
Do you think, MrOllie, the summing up of different meanings is wrong?
Do you think, Sławomir Biały, the summing up of different meanings is wrong?
One of the main reasons I see the present text as ready for improvement, I described earlier in
"It" is NOT a number.
"It" is NOT a sequence (a mapping on N)
"It" is NOT an expression (for the present text says: "a series is represented by an expression)
"It" is NOT a function.
"It" is 'associated' (what's that?) with a sequence. "It" is sometimes 'associated' with a value.
"It" has terms and partial sums.
"It" can have a limit, a value, a sum.
"It" can be geometric, arithmetic, harmonic, alterating, converging, diverging, absolute converging, and more.
What's in fact the content of this black "it"-box? It seems to be empty.
I'm going to replace this unsatisfactory text by an alternative introduction. Chiefly identical with what was shown in this Talk page
here, 18 April 2017. The only reaction on it was the remark that "Hesselp doesn't understand what A SERIES IS (in mathematics)". I agree with that. --
Hesselp (
talk) 22:05, 24 April 2017 (UTC)
--
1. (Sent.1) "a series IS ... the sum of the terms of ..."
Being the sum of numbers again a number, the words 'series' and 'number' are used as synonym. A few lines later it is said that this is not intended.
2. (Sent.2) "a series continues indefinitely"
What is meant by: an indefinitely continuing 'sum of the terms of something' ?
3. (Sent.4) "the value of a series"
What is meant by: the value of a sum (a number) ?
4. (Sent.4) "evaluation of a limit of something"
What's meant with this?
Is it true that a series doesn't have a value, without that limit being 'evaluated' ?
Is it always possible to 'evaluate' the limit of a sequence of terms ?
5. (Sent.5) "the expression obtained by adding all those (an infinite number of) terms together"
A (symbolic, written) expression can be obtained by writing down some symbols using a pen or pencil (or using the keys of a keyboard). The task of adding an infinite number of terms is not feasible, so never any expression will be obtained.
6. (Sent.6) "obtained by placing the terms side-by-side with pluses in between them.
This 'placing' sounds much better feasible. I miss the three centered dots ('bullets') at the right end.
7. (Sent.6) "infinite expression"
I see 'series' and 'infinite sum' used as synonyms for 'infinite expression'. But what notion / mathematical object is denoted by this labels ? It must be a notion 'not being a part of the conventional foundations of mathematics'. How many readers of this article are acquainted with this notion already by themselves?
8. (Sent.7) "The infinite expression can be denoted ..." Such expressions mostly denote a number, a function or a sequence. But an expression denoting a expression sound very strange.
9. (Sent.9) "two series of the same type"
I cannot find where is explained what is meant by: 'the type of that mysterious notion called series '.
10. (Sent. 8, 9, 10, 11, 12)
Is the (intended) information communicated by this five sentences really of enough importance to be incorporated in the 'introduction' ?
11. (First line after 'Definition') The twofold description of the meaning of the word 'series' (as sum, and as expression) causes - unnecessary? - complexity.
--
@Slawomir. Never in my life I've denied that mathematical expressions are totally different from numbers. You must have misunderstood me somewhere, I cannot trace back where this could have happened.
I agree with you on everything you wrote in the first 7 sentences in 12:46, 2 May 2017(UTC) (Until "The sigma notation for a series..."). About your sentences 8, 9, 10 I'm not sure.
Maybe things become more clear from your judgment of the following statements a - h (true or false):
a) the expression e+π evaluates to (= has as its value) the number e+π
b) the expression 1+1 evaluates to the number 1+1
c) the expression 1+1 evaluates to the number 2
d) the sigma expression Σi =1∞ ai evaluates to the infinite expression a1+a2+a3+···
e) Provided that limn→∞ (a1+ ··· +an) exists,
in other words limn→∞ (a1+ ··· +an) is a valid expression,
in other words sequence (an) is summable,
the infinite expression a1+a2+a3+··· (number-interpretation) evaluates to the number limn→∞ (a1+ ··· +an)
f) the infinite expression a1+a2+a3+··· (sequence-interpretation) evaluates to the sequence (a1+ ··· +an)n≥1
g) Being p1, p2, p3, ··· successive primes,
the infinite expression p1-3+ p2-3 + p3-3+ ··· evaluates to the number p1-3+ p2-3 + p3-3+ ···
h) the infinite expression 9− 9^1+ 9− 9^2+ 9− 9^3+ ··· evaluates to the number Σi =1∞ 9− 9^í
According to me this is a quite peculiar way to use the verb 'to evaluate' (in the intro of the present text: "A series is thus evaluated by examining ...."); you can show sources? I only saw it, meaning: given an expression (denoting a number), find the decimal representation of its value, exact or approximated. --
The intro of the present text explains the meaning of "series" using:
The series of an given infinite sequence is the infinite expression that is obtained by placing terms side-by-side with pluses in between.
By 'infinite expression' is not meant an expression with infinite physical dimensions. Nor an expression of the type "1/0".
The
Wikipedia article says: "an expression in which some operators take an infinite number of arguments". That's sufficiently clear to most of our readers? I doubt
Moreover, that article has: "Examples of well-defined infinite expressions include infinite sums, whether expressed using summation notation or as an infinite series, ....". With a circulating reasoning, because 'infinite sum' is linked to the article named ....'Series (mathematics)'. --
In the discussion on the concept/object/idea/entity (mathematical or philosophical) named "series", a number of negative statements are made on this Talk page, since April 10, 2017. Two new ones are found in a post by Sławomir Biały,
dated 21:57, 2 May 2017(UTC): - "it" is NOT a numeral - "it" is NOT an expression that denotes a number.
"It" is NOT a sequence (a mapping on N)
"It" is NOT an expression
"It" is NOT a function
"It" is NOT a part of Zermelo-Fraenkel set theory
"It" is NOT a part of the conventional foundations of mathematics.
"It" is NOT a numeral
"It" is NOT an expression that denotes a number
"It" is 'associated' (what's that?) with a sequence.
"It" has terms and partial sums.
"It" can have a limit, a value, a sum.
"It" is represented by an expression
"It" is sometimes 'associated' with a value.
"It" can be geometric, arithmetic, harmonic, alterating, converging, diverging, absolute converging, and more.
--
Bovenstaande is (nog) niet geplaatst!
On edit 15:34, 4 May 2017.
What is meant:
a series is a description of the operation: adding one-by-one infinitely many quantities
or
a series is the operation: adding one-by-one infinitely many terms ?
What a reader should think of: an operation that cannot be carried on (not 'effectively') ?
I'm curious to see how you define (based on reliable sources): "a convergent infinite adding operation", "a alternating infinite adding operation" "a geometric infinite adding operation" "a Fourier infinite adding operation" "the Cauchy product of two infinite adding operations" "a power infinite adding operation" and much more.
Please present a mature proposal for the intro-plus-definition part of the article. Here on Talk page, so not unnecessary disturbing our Wiki-readers . --
In Talk page, no user took part in discussion on the merits of the content of this text. So 'no consensus' cannot be a valid reason to revert.
From the 'edit summary' 13:22, 5 May 2017: "...most editors have already given up trying to communicate with you" .
That 'trying to communicate' refers to reactions with no more relation to the content of the proposed text-section, than in phrases of the type:
- don't agree with proposed changes - undocumented POV-pushing - Hesslp doesn't understand what a series is - this talk page is not for discussing personal opinions about the practice of mathematicians - this is not mathematics, it is philosophy - you have clearly a misconception of what is mathematics - for being clearer, every line of Hesselp's post is either wrong, or does not belong to this talk page or both - I reiterate my objection .
Attempts made to start discussion, in this list:
"It" is NOT a number.
"It" is NOT a sequence ( a mapping on N)
"It" is NOT an expression
"It" is NOT a function.
"It" is NOT a part of Zermelo-Fraenkel set theory
"It" is NOT an expression that denotes a number
"It" is NOT a numeral
"It" is represented by an expression
"It" is 'associated' (what's that?) with a sequence.
"It" is sometimes 'associated' with a value.
"It" has terms and partial sums.
"It" can have a limit, a value, a sum.
"It" can be geometric, arithmetic, harmonic, alterating, converging, diverging, absolute converging, and more.
1. (Intro, sentence 1) "a series IS ... the sum of the terms of ..."
(Being the sum of numbers again a number, the words 'series' and 'number' are used as synonym.)
2. (Intro, sent.5) "The series of (associated with) a given sequence a IS the expression a1+a2+a3+··· "
(The word 'series' used as the name of a mapping.)
3. (Definition, sent.1) "a series IS an infinite sum, which is represented by a written symbolic expression of a certain type."
(It isn't clear whether or not the clause after the comma is part of the definition. 'IS' a series still an infinite sum, in situations where it is not represented by an expression of the intended form?)
4. (Definition, sent.6) "series(pl) ARE elements of a total algebra of a ring over the monoid of natural numbers over the a commutative ring of the a's "
(The word 'series' as the name for elements of a certain structure; just as the word 'number' is used as the name for elements of another mathematical structure. To which element in this 'definition' is referred by "the a's" ? )
In case it is accepted that the word 'series' has four different meanings in mathematics (is used in four different ways) the first part of the article headed by "Series" should be structured like:
a. The word 'series' is used as name/label for ......... .
b. The word 'series' is also used as name/label for ......... .
c. The word 'series' is used as name/label for .......... as well.
d. Moreover, sometimes the word 'series' is used as name/label for ......... .
The present text directs the reader to believe that there is ONE and only ONE sacred given-by-God-meaning of this word.
That's religion, not mathematics.
Do you think, Wcherowi, the summing up of different meanings is wrong?
Do you think, D.Lazard, the summing up of different meanings is wrong?
Do you think, MrOllie, the summing up of different meanings is wrong?
Do you think, Sławomir Biały, the summing up of different meanings is wrong?
Being the sum of numbers again a number, the words 'series' and 'number' are used as synonym. A few lines later it is said that this is not intended.
2. (Sent.2) "a series continues indefinitely"
What is meant by: an indefinitely continuing 'sum of the terms of something' ?
3. (Sent.4) "the value of a series"
What is meant by: the value of a sum (a number) ?
4. (Sent.4) "evaluation of a limit of something"
What's meant with this?
Is it true that a series doesn't have a value, without that limit being 'evaluated' ?
Is it always possible to 'evaluate' the limit of a sequence of terms ?
5. (Sent.5) "the expression obtained by adding all those (an infinite number of) terms together"
A (symbolic, written) expression can be obtained by writing down some symbols using a pen or pencil (or using the keys of a keyboard). The task of adding an infinite number of terms is not feasible, so never any expression will be obtained.
6. (Sent.6) "obtained by placing the terms side-by-side with pluses in between them.
This 'placing' sounds much better feasible. I miss the three centered dots ('bullets') at the right end.
7. (Sent.6) "infinite expression"
I see 'series' and 'infinite sum' used as synonyms for 'infinite expression'. But what notion / mathematical object is denoted by this labels ? It must be a notion 'not being a part of the conventional foundations of mathematics'. How many readers of this article are acquainted with this notion already by themselves?
8. (Sent.7) "The infinite expression can be denoted ..." Such expressions mostly denote a number, a function or a sequence. But an expression denoting a expression sound very strange.
9. (Sent.9) "two series of the same type"
I cannot find where is explained what is meant by: 'the type of that mysterious notion called series '.
10. (Sent. 8, 9, 10, 11, 12)
Is the (intended) information communicated by this five sentences really of enough importance to be incorporated in the 'introduction' ?
I agree with you on everything you wrote in the first seven sentences in 12:46, 2 May 2017(UTC) (Until "The sigma notation for a series..."). About your sentences 8, 9, 10 I'm not sure. Maybe things become more clear from your judgment of the following statements a - h (true or false, or ...):
a) the expression e+π evaluates to (has as its value) the number e+π
b) the expression 1+1 evaluates to the number 1+1
c) the expression 1+1 evaluates to the number 2
d) the sigma expression Σi 1∞ ai evaluates to the infinite expression a1+a2+a3+···
e) Provided that limn→∞ (a1+ ··· +an) exists,
in other words limn→∞ (a1+ ··· +an) is a valid expression,
in other words sequence (an) is summable,
the infinite expression a1+a2+a3+··· (number-interpretation) evaluates to the number limn→∞ (a1+ ··· +an)
f) the infinite expression a1+a2+a3+··· (sequence-interpretation) evaluates to the sequence (a1+ ··· +an)n≥1
g) Being p1, p2, p3, ··· successive primes,
the infinite expression p1-3+ p2-3 + p3-3+ ··· evaluates to the number p1-3+ p2-3 + p3-3+ ···
h) the infinite expression 9− 9^1+ 9− 9^2+ 9− 9^3+ ··· evaluates to the number Σi 1∞ 9− 9^í
According to me this is a quite peculiar way to use the verb 'to evaluate' (in the intro of the present text: "A series is thus evaluated by examining ...."); you can show sources? I only saw it, meaning: given an expression (denoting a number), find the decimal representation of its value, exact or approximated. -- Hesselp ( talk) 21:37, 2 May 2017 (UTC)--
There is a situation with Hesselp ( talk · contribs · deleted contribs · logs · filter log · block user · block log) on the page Series (mathematics) and the talk page Talk:Series (mathematics). He has been edit-warring to include his rewrite of the article [3], [4], [5], [6], [7], [8]. Although not at the moment above 3RR, the above is clear indication of edit warring, being reverted by four different editors. He was warned against edit warring, yet persists. Other editors have attempted to engage him at Talk:Series (mathematics), but attempts to resolve the dispute amicably are met with walls of antagonistic rambling text: [9], [10], [11], [12], [13], among others. We have given up on trying to interact with this user, in the spirit of WP:DENY (the above posts strongly suggest trolling). But I believe the time has come for this disruption to be put to an end administratively. (Pinging other involved editors: @ Hesselp:, @ D.Lazard:, @ MrOllie:, @ Wcherowi:.) Sławomir Biały ( talk) 11:58, 6 May 2017 (UTC)
Toegevoegd vraag aan L3X1 (?)
D.Lazard writes (15:14, 30 April 2017): "Presently, mathematicians agree on the concept of a series, but as usual for concepts that have many applications, the formal rigorous definition is too technical for being understood by beginners, .....".
This 'agree on' seems to be not in accordance with the ongoing rewriting of the Definition section in the article. Not with the absence of a decisive unambiguous source. And not with the result of a survey, made around 2008. About eighty books on calculus were inspected, the results are shown below (press [show]). The original language was not always English; capital-sigma forms were seen as not different from a1 + a2 + a3 + ··· .
Bowman, Britton/Kriegh/Rutland, Edwards/Penny, Open University-UK, Small/Hosack
2. An (infinite) series is an expression that can be written in the form a1 + a2 + a3 + ···
Anton/Herr, Anton, Anton/Bivens/Davis
3. An (infinite) series is a formal sum of infinitely many terms.
R A Adams
4. An (infinite) series is a formal infinite sum.
Ahlfors
5. The formal expression a1 + a2 + a3 + ··· is called an (infinite) series.
Matthews/Howell, Sherwood/Taylor
6. An (infinite) series is an indicated sum of the form a1 + a2 + a3 + ···
Kaplan
7. An (infinite) series is a sequence a1, a1 + a2, a1 + a2 + a3, ···
Hurley
8. An (infinite) series is a sequence whose terms are to be added up.
Marsden/Weinstein
9. An (infinite) series is the indicated sum of the terms of a sequence.
Daintith/Nelson, Kells, Weber
10. An (infinite) series is the sum of the terms of a sequence.
Wikipedia-Spanish
11. An (infinite) series is the sum of a sequence of terms.
Borowski/Borwein
12. An (infinite) series is the sum of an infinite number of terms.
Lyusternik/Yanpol'skii
13. An (infinite) series is a sum of a countable number of terms.
Borden
14. An (infinite) series is an infinite addition of numbers.
Goldstein/D C Lay/Schneider(/Asmar)
15. An (infinite) series is an infinite sum: a mathematical proces which calls for an infinite number of additions.
Davis/Hersh
16. An (infinite) series is a sequence of numbers with plus signs between these numbers.
Bers
17. We have an (infinite) series if, between each two terms of an infinite sequence, we insert a plus sign.
Maak
18. An (infinite) series is an ordered pair {an}; {sn} with sn short for a1 + a2 + … + an
Buck, Gaughan, Maurin, Protter/Morrey, Zamansky, Encyclopaedia of Mathematics1992,
Wikipedia-Dutch, Wikipedia-English, Wikipedia-French
Buck writes(1956,1965, 1978): "An infinite series is often defined to be 'an expression of the form Σ1∞ an '. It is recognised that this has many defects."
19. If we try to add the terms of an infinite sequence a we get an expression of the form a1 + a2 + a3 + ··· which is called an (infinite) series.
Stewart
20. If we add all the terms of an infinite sequence, we get an (infinite) series.
De Gee
21. When the terms of a sequence are added, we obtain an (infinite) series.
Croft/Davison
22. When we wish to find the sum of an infinite sequence <an> we call it an (infinite) series and write it in the form
a1 + a2 + a3 + ···
Keisler
23. Given a sequence a , then the sequence a1, a1 + a2, a1 + a2 + a3, ··· is called an (infinite) series.
Apostol, Burrill/Knudsen, Endl/Luh, Fischer, Forster, S R Lay, Rosenlicht, Wikipedia-Italian
24. Given a sequence a, then the sequence a1, a1 + a2, a1 + a2 + a3, ··· is called the (infinite) series
connected with the sequence a.
Barner/Flohr, Friedemann,
Dijkstra cs (Twente University), Almering (Delft University)
25. Given a sequence a, then the infinite sum a1 + a2 + a3 + ··· is called an (infinite) series.
Grossman, Leithold
26. Given a sequence a, then the expression a1 + a2 + a3 + ··· is called an (infinite) series.
L J Adams/White, Blatter, Van der Blij/Van Thiel, Gottwald/Kästner/Rudolph, Sze-Tsen Hu
27. Given a sequence a, the symbolic expression a1 + a2 + a3 + ··· we call an (infinite) series.
Rudin, Walter
28. Given a sequence a, an expression of the form a1 + a2 + a3 + ··· is an (infinite) series.
Thomas/Finney
29. No explicite attempt is made to describe the meaning of (infinite) series, although this term is used frequently.
Ackermans/Van Lint, Binmore, Cheney, Godement, Hille, Hirschman, Johnson/Kiokemeister, Knapp, Kreyszig, Larson/Hostetler, Lax, Morrill, Neill/Shuard, Riley/Hobson/Bence, Van Rootselaar, Ross, Varberg/Purcell/Rigden, Widder, Wikipedia-German, Duistermaat (Utrecht University), D&I (Groningen University)
30. For any sequence , the associated (infinite) series is defined as the formal sum (expression describing a sum) aM + aM+1 + aM+2 + ··· .
Wikipedia-Dutch (fall 2015)
31. An infinite sequence of real numbers is called (infinite) series. Original wording: On appelle 'série' une suite indéfinie de quantités (quantité: nombre reel).
C.-A. Cauchy.
This not very satisfactory situation, caused by the double meaning of 'convergence' in the 19th century, can be structured by accepting that:
- when 'series' is used denoting a mathematical object, it is synonym with 'sequence' (as in the 19th century and later), and
- in other cases 'series' is designating a certain kind/type of expression (or representation, or evaluation, or maybe even more).
Instead of 'series expression' mostly the shorter 'series' is used. But one has to realize that with 'convergent series' is not meant: 'the convergent mathematical object named series ', but: the convergent mathematical object denoted by the (type series) expression.
--
@Sławomir Biały. Please, present one or more explicit examples of occurrences of "antagonistic text" in my posts on this Talk page. And one or more examples of occurrences of "rambling text" in my posts on Talk page.
I hope I can learn from your examples, how to improve the presentation of my arguments. And how to avoid unnecessary blocking. --
To the list of "Secondary sources supporting Hesselp's edits" (22:52, 27 April 2017, answering Wcherowi's remark 17:16, 25 April 2017 "...your edits are not supported by citations to reliable secondary sources...") I add:
-
R. Creighton Buck (1920-1998, University of Wisconsin), Advanced Calculus, 1st ed. 1956, 2nd ed. 1965, 3rd ed. 1978
"An infinite series is often defined to be 'an expression of the form Σ1∞ an '. It is recognised that this has many defects."
--
Spivak: editions 1967, 1980, 1994, 20??(nog controleren)
"........an acceptable definition of the sum of a sequence should contain, as an essential component, terminology which distinguishes sequences for which sums can be defined from less fortunate sequences."
D.Lazard, 15:14, 30 April 2017(UTC)
...a series is a
mathematical object. It appears that this concept is not a simple one, as it involves the concept of
infinity, which was not well understood nor well accepted before the end of the 19th century (this make your citation of Cauchy irrelevant for discussing the modern terminology; note that he avoided carefully to talk about infinity).
Wikipedia "Mathematical object": A mathematical object is an abstract object arising in mathematics. .... In mathematical practice, an object is anything that has been (or could be) formally defined, ...
Victor J. Katz, A History of Mathematics An Introduction (reprint November 1998), p.705
It was Augustin-Louis Cauchy, the most prolific mathematician of the nineteenth century, who first established the calculus on the basis of the limit concept so familiar today. Although the notion of limits has been discussed much earlier, even by Newton, Cauchy was the first to translate the somewahat vague notion of a function approaching a particular value into arithmetic terms by means of which one could actually prove the existence of limits. Cauchy used his notion of limit in defining continuity (in the modern sense) and convergence of sequences, both for numbers and of functions. .......
I have no desire to enter long discussions about this article, but I wanted to leave a few comments about this revision [14]:
— Carl ( CBM · talk) 15:44, 8 May 2017 (UTC)
@Carl. Thank you very much for your concrete comments.
On point 1: I understand your remark. But......in this case? You add: "whenever possible". Here we have a mathematical object: (in modern words) a mapping on N. The traditional word for what later on is normally named "sequence". And we have a mathematical concept(?), a certain type of expression (a sign for the 'infinite summation function' plus a sign for a sequence as its argument). You may change the order of the two. The same 'series-type' we meet when classifying representations (for numbers or functions), and when classifying expansions (for functions).
I'm afraid this cannot be combined in one phrase. I explained this in my article text.
On point 2: I plan to smooth the content of this footnote. Maybe omit it completely. You are right that this sharp, maybe exaggerated wording is better suited for a discussion on Talk page.
On point 3: On the unusual spacing in R e d u c t i o n o f . . . you're 100% right, I was lazy when I copied it from elsewhere. On the use of other extra spacings: you cannot see them as making the text, and the formulas, better readable? Enough to accept some deviation from standard style?
And on the use of more 'ordinary prose': maybe a question of taste as well. I shall reconsider this. I wouldn't take as an example the present text of the article. For me that's very far from any encyclopedic style. --
The facts, in short: David Eppstein was 'baffled' (Talk page 22:01, 8 May 2017) by my incomprehension regarding the true nature of "expressions" and "infinite expressions" (being the central key-term in the definition of 'series'). After asking for the difference between finite and infinite expressions (09:38, 9 May, again 08:44, 10 May), the answer (14:36 and 15:43) was unclear to me, so I made my question more concrete (points A-E, 18:49, 10 May). Reaction by David Eppstein: "...no more interaction with you", "I see your edits as tendentious and disruptive" and some more not very positive remarks. --
D.Lazard's post in WP:ANI is copied here, in three parts with comments by Hesselp indented.
Hesselp's version of
series (mathematics) begins by "In mathematics (calculus), the word series is primarily used as adjective ...
". This is not only
WP:OR but also blatantly wrong: It suffices to look at any modern textbook of calculus to know that "series" is primarily used in mathematics as a noun.
Note also that, although series are studied in most textbooks of calculus the only source for Hesselp's lead is about 150 years old (and also misunderstood).
The remainder of Hesselp's version of the article continues in the same style and consists only of Hesselp's own inventions, beliefs and/or misinterpretation of the rare source that he produces. D.Lazard ( talk) 23:35, 10 May 2017 (UTC
Now we observe another attack toward " Series (mathematics)" (see "Relevant discussion at WP:ANI" above); User:Hesselp insists on a single definition of a series as a sequence (of terms). For now the article defines a series as (a special case of) an infinite expression. Another equivalent definition in use is, a pair of sequences (terms, and partial sums). Regretfully, this case is not covered by my "bastion", since the set of series is itself not quite an instance of a well-known mathematical structure (though some useful structures on this set are mentioned in our article). And still, it would be useful to write something like A person acquainted with series knows basic relations between terms and partial sums, and does not need to know that some of these notions are "primary", stipulated in the definition of a series, while others are "secondary", characterized in terms of "primary" notions. Implementation need not be unique. When several implementations are in use, should we choose one? or mention them all "with due weight"? or what? Any opinion? Boris Tsirelson ( talk) 16:40, 12 May 2017 (UTC)
The same is the case with the 'definition' in the present version of the article; implying the self-referring sentence: A series is an expression of the form ..+..+..+ ··· denoting a series.This seems a quote, but the word "denoting" does not appear in the article. This method of changing the wording of the content that he pretends discussing is systematic. This strongly suggests a bad faith; in any case it is definitively impossible to have a constructive discussion with this editor. Therefore, a permanent ban seems the only acceptable solution. D.Lazard ( talk) 21:18, 12 May 2017 (UTC)
The question whether or not a sound criterion exists to decide between finite expression and infinite expression is mentioned in the following posts:
21:50 2 May 2017,
09:38, 9 May 2017,
15:43, 10 May 2017,
18:49, 10 May 2017,
20:45, 10 May 2017,
22:19, 10 May2017.
No clear answer on this question is formulated yet.
(moet nog afgemaakt)
In mathematics (calculus), the word series is primarily". This is not only WP:OR but also blatantly wrong:"
used as adjective ...
The same is the case with the 'definition' in the present version of the article; implying the self-referring sentence: A series is an expression of the form ..+..+..+ ··· denoting a series.This seems a quote, but the word "denoting" does not appear in the article.
Comments on D.Lazard's post 23:35, 10 May 2017 :
On "..any modern textbook.." : For a survey of attempts to define 'series', see the list '32 attempts' in
this post. The 32 different wordings can be combined to a handful of really different content. Most of the about 80 authors say that a series IS an expression, but leave it to the reader to find out what's the character of the mathematical object, denoted (described, referred to) by this expression. The same is the case with the 'definition' in the present version of the article; implying the self-referring sentence: A series is an expression of the form ..+..+..+ ··· denoting a series.
And left to the reader as well is the question of how to interprete the label "convergent series". A convergent expression seems to be nonsense, but without any idea about the content of the expression, it's not easy to understand what's really denoted by this label.
In some sources (
Spivak,
Buck,
Dijksterhuis,
Van Rooij,
Cauchy,
Gauss) can be found more explicitely how to interprete the usual wordings. Making it possible to see the connection between the traditional - self-referring - wordings in most books on calculus, and the way how the label 'series' is used by mathematicians in practice.
Only a minimal change in interpretation is needed. That is: don't say: 'series' IS the expression ..+..+..+ ··· itself, but say: 'series' is used to label a certain TYPE OF expression. The type, constituted by a summation symbol (the sigma-sign, or the repeated pluses and end-dots) combined with the name of a sequence.
This is what should be an improvement of the article, with its consequences in the wording of the remaining standard content. Helping the reader to grab the meaning of the on-first-site strange combination 'convergent series' (= convergent expression).
Original Research ?: The explanation of the meaning of 'convergent series' - as being nothing else as summable sequence - is the very first statement in chapter 'Series' in Michael Spivak's well known "Calculus". Already for half a century: 1st ed. 1967, 4th ed. 2008. See More precise terminology 21:37, 9 May 2017
"The only source...."? No, all 80 rather modern calculus books in the list in this post, 20:28, 8 May 2017 served as sources. And of the 19th century sources are mentioned earlier: Cauchy, Susler, Itzigsohn, Gauss, Von Mangoldt. Why doesn’t D.Lazard mentions which one of this five he has studied, and which point in it I should have misunderstood?
The remainder of Hesselp's version.... Without concrete examples, I can't comment on D.Lazard's last sentence. Is it the conclusion of everyone who have read this edit? --
Attempting to find a way to some kind of consensus, I add the following lines to this Talk page.
Citations, taken out of longer posts on
Wikipedia talk:WikiProject Mathematics
-
Tsirel - 19:15, 12 May 2017: ".. in general an expression has no value (but in "good" cases it has);" (Comment Hesselp: the dispute is about the question whether a series-type expression has (in "good" cases) a number as its value, or a series (For: "a series is denoted by an expression like ..+..+..+···"))
-
CBM - 20:00, 12 May 2017: "... the definitions that are often given in the books lack something that would be present in a graduate level text." (Comment Hesselp: No one has presented such a graduate level text in this Talk page.)
-
CBM - 20:00, 12 May 2017: "...we should follow the sources and present the same general understanding that they convey.] (Comment Hesselp: That's easier said than done, see survey in
09:38, 9 May 2017)
-
CBM - 20:09, 12 May 2017: "If numerous sources all find it possible to discuss a concept without a formal definition, we can certainly do so as well."
-
D.Lazard - 20:43, 12 May 2017: " In any case, a series is not a sequence nor a pair of sequences nor an expression. It is an object which is built from a sequence." (Comment Hesselp: D.Lazard's edited since
09:50, 14 Februari 2017 seven times a version with: "a series is an expression").
-
Tsirel - 05:02, 13 May 2017: "What does it mean? A vague term whose meaning is determined implicitly by the context, case-by-case?"
-
Taku - 23:10, 13 May 2017: "... a series is a more of a heuristic concept than an explicitly defined concept."
Observations Studying the terminology used in the 19th (and a good part of the 20th) century, concerning the 'series-representation' of numbers (and of functions), we can see two noteworthy points.
(1) The word 'series' was used frequently in situations where we should use 'sequence' now. (Also German 'Reihe' in 'Folge'-situations, and French 'série' in 'suite'-situations.) Cauchy introduces 'série' explicitely for a sequence with numbers as terms; much later Bourbaki seems to copy this by using 'series' for a sequence with terms allowing the existence of a 'sum series'. The names 'arithmetical series', 'harmonical series', 'Fibonacci series', etc. were in common use. (2) The words converge/convergent/convergence were used in case the terms have a limit, as well as in case the partial sums have a limit. Cauchy seems to use the verb 'converger' for terms with a limit, and the adverb 'convergent' for partial sums with a limit; quite confusing. And Gauss once remarks: (Werke Abt.I, Band X, S.400) "Die Convergenz einer Reihe an sich ist also wohl zu unterscheiden von der Convergenz ihrer Summirung ...." (The convergence of the sequence itself has to be distinguished from the convergence of its summation.)
Suppostion This situation: two words (series and sequence) for one notion, and one word for two properties (limiting terms and limiting partial sums), caused ongoing confusion. More and more culminating in a belief in the existence of a third 'mathematical object', apart from 'sequence' and 'the sum sequence of a given sequence'. A mysterious object or notion, whose definition/description causes the difficulties mentioned in the citations above.
How about the idea of describing this historical roots of the present confusion, in the Wikipedia article? Can this be seen as a description of the existing situation, or is this seen as OR? --
I don't think the article should focus on the historical roots to any great extent, except perhaps in a section on history. Sources from the 19th century are not likely to be of much use in this kind of elementary article, and indeed there were many more terminological problems at that time (compare the common use of "infinitesimal" at that time). Every contemporary calculus book I have seen has the same concept of a series, although of course the wording may vary from one author to another. — Carl (CBM · talk) 01:21, 15 May 2017 (UTC)
- Instead of the heading "Definition", I have in mind: "Names and notations".
- About recent changes in the text of the article:
• The self-referring "A series is an expression denoting a series" can't be found in the text any longer. Improvement.
• In the definition of 'series', the two-track construction "a series is an infinite sum, is an infinite expression of the form .." disappeared. Improvement.
• The "such as" regarding the capital-sigma notation. Improvement. (Maybe some more variants can be shown? As well as
a1 + a2 + ... + an + ... as variant of the pluses-bullets form.)
• The label "infinite expression" (instead of "expression") is still there. Although no criterion is found for decerning. See
, , .
• The intro (almost at the end) says: "When this limit exists, one says that the series is convergent or summable, and the limit is called the sum of the series. And the present definition says: "a series is an infinite sum,..". Combined we get wordings as: "a summable infinite sum" and "the sum of an infinite sum".
I know there are books where you can find this; but it's not very nice and comprehensible. Is it definitely OR to add that it's not unusual to say "summable sequence" and "sum of a sequence" as well? I referred to Spivak (1956...2008) and many hits in Google.
The third sentence in the present text says: "Series are used in most areas of mathematics,..". Isn't it true that the content of this sentence can be worded as well by: "Capital-sigma expressions and pluses-bullets expressions are used in most areas of mathematics".
Why are this notations so important? Because they express a method to denote/describe irrational numbers (and as an generalization also functions) by means of a regular-patterned sequence with more familiar rationals as terms (or 'easier' functions).
The usual word for such a method to describe mathematical objects by means of simpler objects, is "representation". We have: the decimal representation, the continued fraction representation, the infinite product representation, and some more. Not the least important is, what could be called "the infinite sum representation" or - in honour of the famous term - "the series representation". The representation based on the summation function for infinite sequences.
So, instead of saying "series are important" (with the hard to define term 'series'), you could say "the series representation is important" (describable without mysterious words). Is this a so big change that you are going to react with: "impossible, clear OR" ?
Last remark. Caused by personal circumstances I've to tell that I leave by now Wikipedia for at least a couple of weeks. I wish you fruitful discussions. Hessel Pot --