Your recent editing history shows that you are currently engaged in an edit war. To resolve the content dispute, please do not revert or change the edits of others when you are reverted. Instead of reverting, please use the talk page to work toward making a version that represents consensus among editors. The best practice at this stage is to discuss, not edit-war. See BRD for how this is done. If discussions reach an impasse, you can then post a request for help at a relevant noticeboard or seek dispute resolution. In some cases, you may wish to request temporary page protection.
Being involved in an edit war can result in your being blocked from editing—especially if you violate the three-revert rule, which states that an editor must not perform more than three reverts on a single page within a 24-hour period. Undoing another editor's work—whether in whole or in part, whether involving the same or different material each time—counts as a revert. Also keep in mind that while violating the three-revert rule often leads to a block, you can still be blocked for edit warring—even if you don't violate the three-revert rule—should your behavior indicate that you intend to continue reverting repeatedly. MrOllie ( talk) 21:37, 28 April 2017 (UTC)
There is currently a discussion at Wikipedia:Administrators' noticeboard/Incidents regarding an issue with which you may have been involved. Sławomir Biały ( talk) 11:58, 6 May 2017 (UTC)
Oops... you'd better not violate 3RR. Boris Tsirelson ( talk) 20:20, 12 May 2017 (UTC)
Hi. I'm just informing you that following the ANI discussion, the consensus by the community is that you be Topic baned for a period of six (6) months starting today, from editing the article at Series (mathematics) and its talk page. Any infringement of this ban will result in your account being blocked without warning for a duration of the administrator's discretion. This sanction is imposed in my capacity as an uninvolved administrator. You may appeal this sanction at the administrators' noticeboard. You may also appeal directly to me (on my talk page), before or instead of appealing to the noticeboard. The ban remains in force however, until any official change is pronounced. Kudpung กุดผึ้ง ( talk) 10:35, 28 May 2017 (UTC)
Hi, Hessel Pot. Mathematically, I like the point of view pushed by you. I happen to teach analysis only in the second year. But if I taught it in the first year, maybe I would follow you, saying "summable sequence" and "series representation", never just "series".
However, note a difference: on my courses I am the decision maker; here on Wikipedia I am not. Here a point of view cannot be presented until/unless it is widely used. And if it is, it must be presented with "due weight". Boris Tsirelson ( talk) 18:52, 28 May 2017 (UTC)
(Unindent) To
Boris Tsirelson. About the past and the present:
On the Dutch WP the dominant opinion is, that by 'the harmonic series' another mathematical concept is meant than by 'the harmonic sequence'; that the two names are not interchangeable. Why? Because the way the are notated is different (terms separated by comma's vs. pluses). My contra-argument that - at least up to my school-years - the word 'sequence' ('rij' in Dutch) was almost non-existant, is rejected as not relevant now. (About Fibonacci sequence versus Fibonacci series we didn't discuss.)
I added 'calculatable' (a spontaneous invention, as a variant on your 'writable') after I saw Swokowski's sentence (p. 533): "To calculate S5, S6, S7, and so on, we add more terms of the series.". With the word 'calculate' he suggest an action/activity (comparable with the 'addition-action' in primary school), which is not possible to perform in the general case. Imo Sw. leads the reader into a wrong direction with this 'calculate'.
The same wrong direction is suggested (imo) by the use of 'infinite sum' in the key-sentence of the series-article. The existence of a mathematical concept named 'infinite sum' is suggested, being the result of some kind of calculating/working with the given terms.
I did not at all mean 'countable number'. --
Hesselp (
talk) 23:18, 26 June 2017 (UTC)
- I. Are you aware of the fact (is it a fact?) that the word 'series' is very often used in direct connection with (mostly directly followed by) a symbolic, written form of type 'capital-sigma' or type 'pluses-bullets'
( Σn=1∞ or or variants).
- II. As a consequence of I, the word 'series' will be absent in the oral/verbal part of your lectures on calculus (without a blackboard or a more modern pictural divice for communication). I don't have a verbal alternative for a capital-sigma-form or a pluses-bullets-form. --
Hesselp (
talk) 21:43, 28 June 2017 (UTC)
- III. Can you imagine that I see the facts described in I and II as a strong indication that the word 'series' is not in use for a mathematical concept, but for a notational form?
- IV. With your three definition-variants (1, 1', and 1") you want to illustrate that, in your view/interpretation, the terms-sequence (an) and the sums-sequence (Sn) have an equivalent role in the concept/object 'series'. Right? I paraphrase your triple by:
"Definition: An ordered pair of real-number-sequences an; bn , related by bn = a1 + ··· + an (or the equivalent: a1 = b1, an+1 = bn +1 − bn ), is called a (real-number-)series." Right?
- V. Why this emphasis on the equivalent role of both sequences? The relation between a (summable) terms-sequence and the limit of its sums-sequence is an important one in calculus. As is the relation between a (complicated) function and the sequence of its (less complicated) Maclaurin-terms. As well as the relation between f and the sequence of the Fourier-terms of f. On the contrary, the relation between the sums-sequence and the limit of its terms-sequence has no applications in calculus (as far as I know). So I don't see a good reason to emphasize an equivalency of terms-sequence and sums-sequence. Right?
- VI. I've another argument against adopting the pair of sequences as being the heart of a 'series'-concept. Isn't it standard use in mathematics to formulate definitions as simple/elementary as possible? So why should the sums-sequence be mentioned in the definition, while this sums-sequence is already completely determined by the other sequence?
- VII. Your pair-of-sequences as being the mathematical meaning of 'series', I met earlier in calculus-books by: Creighton Buck 1956, Zamansky '58, Apostol '74, Maurin '76, Protter/Morrey '77, Encyclopaedia of Mathematics '92, Gaughan '98, Boos '00, Edward Azoff '05. Not long ago I red that the source is in a Bourbaki-publication. (I cannot find back where I saw this; do you know this source?) As I said in VI, this definition doesn't say anything more than 'series' is another word for 'sequence'. Or more precise: A sequence with a sums-sequence (the same as: a sequence with additionable terms) is called series.
Or can you describe a difference between "a terms-sequence sums-sequence pair" and "a sequence with terms that can be added to form its sums-sequence" (essentially the same as Cauchy's: "une suite de nombres réels") ? --
Hesselp (
talk) 21:43, 28 June 2017 (UTC)
- VIII. Once more on your: "How do I interpret the word "series" when reading or writing an article in a mathematical journal?".
Can you find back places in articles/books written by you (or red by you) where you used/red the word 'series' ? (The articles/books not being tutorial texts on calculus.) Can at that places the word 'series' be red as: "the combination of expressions for the summation function and for a sequence"? Or can "series Σ a " (or "series a1 + a2 + ··· ") be red as "sequence a" ? --
Hesselp (
talk) 05:36, 29 June 2017 (UTC)
(I wrote my 'point VIII' before your '(Unindent)' came in, but was just some minutes late in posting it.)
- IX. On your: "It seems we agreed to put metamathematical notions aside (wherever possible).": I see this put aside as not possible in a discussion on the way mathematicians use the word 'series' in a calculus context. The consequence of this put aside should be (imo) that you want to skip WP-articles with titles: 'sum', 'product', 'quotient', 'logarithm', 'integral', 'root', 'power', et cetera.
- X. On your: "...surely I could pronounce..." and your "at some point I would say [!,HP]...": Do you really say (verbally, in words), with your back to the blackboard: "series Greek capital letter sigma indexed by n is one and by infinite a indexed n " ? Or something like this, starting with the word 'series' ? --
Hesselp (
talk) 07:03, 29 June 2017 (UTC)
Sorry, it is impractical to discuss 8 points in parallel. Let us close one point and then start another.
Boris Tsirelson (
talk) 06:27, 29 June 2017 (UTC)
Yes, it can be practical to discuss points I - X (maybe some combined) in different (sub)sections. --
Hesselp (
talk) 07:03, 29 June 2017 (UTC)
Well, then create the subsections; but anyway I do not want to scatter, and will attend one subsection at a time. The more so that the conclusion reached in one subsection may affect the discussion of another. Thus, choose their order...
Boris Tsirelson (
talk) 07:11, 29 June 2017 (UTC)</math>
I leave it to you, Tsirel, to choose the order. Or else: why not the order of the roman numbers? --
Hesselp (
talk) 08:09, 29 June 2017 (UTC)
OK, I start the split process.
Boris Tsirelson (
talk) 09:14, 29 June 2017 (UTC)
Maybe I understand your point, at last!
Here is my guess. We have here another oddity of mathematical terminology. (Strangely I did not note it before.) First, both notations, and are ambiguous; depending on the context they may denote either a series (that has terms and partial sums), or the sum of this series! Examples:
in (a) the expression denotes the series, but in (b) a similar expression denotes the sum of the series.
Is the word "series" ambiguous in the same way? Probably, less ambiguous, since usually one writes "sum of the series", not just "series", when the sum is meant. But probably a less accurate language is used sometimes, like this:
in this phrase the "series" does not exceed a number, therefore the "series" is interpreted as a number. Though, one may say that no, the author uses element-wise inequality between one series and another series, and deduces inequality between their sums.
Does a similar oddity apply to integrals? Usually, denotes the number (rather than the whole integrand function). But probably a less accurate language is used sometimes, like this:
surely in (d) one does not mean Riemann sums for the number and in (e) one does not mean that a number could be absolutely convergent. Boris Tsirelson ( talk) 18:59, 29 June 2017 (UTC)
@
Tsirel
A. "it is not about contradictions in mathematics [...] but about oddities of mathematical terminology"
A mathematical or a pedagogical problem? I cannot choose. But in the present text of
de:Reihe (Mathematik) I read in Satz 3 that 'Reihe' is the name for every Folge with a Differenzenfolge (e.g. every Zahlenfolge). And in Satz 8 (in a cripple way) that 'Reihe zum Folge (an)' means the same as 'Partialsummenfolge der Folge (an)' . Sätz 3 and 8 are contradictional, imo. At least it should be noticed that this are two different ways the word 'Reihe' is used in calculus.
B. How do first class mathematicians (try to) solve this definition-question? See:
- Bourbaki, Éléments de Mathématique, Première partie, Livre III (Topologie générale), Chap. 3, Par. 4, No 6. Series (Deuxième Édition, 1951, p. 42-43) "On appelle série définie par la suite (xn) le couple des suites (xn) et (sn) ainsi associées."
-
Encyclopedia of Mathematics - Series: "A pair of sequences of complex numbers {an} and {a1+ ··· + an} is called a (simple) series of numbers"
- B. Tsirelson
28 June 2017: "Definition 1. A series (of real numbers) is a pair of (infinite) sequences and (of real numbers) such that for each ."
This definition - though formally not incorrect - is absurd, for you can choose whatever you want as second element in the pair. "Series" can be defined as being an ordered pair with an infinite sequence-with-addition as its first element and a arbitrary object as its second. Consequences:
an alternating/harmonic/Fibonacci series a pair with an alternating/harmonic/Fibonacci sequence as its first element;
a convergent series a pair with as first element a sequence with converging partial sums;
the terms / partial sums of a series the terms / partial sums of its first element
the sum of a series ( (an) ; .... ) the limit of the partial sums of sequence (an).
A series is a pair? Did you ever see (1, 1/2, 1/4, 1/8, ... ; 1, 3/2, 7/4, 15/8, ...) as an example of a convergent series?
C. "Experienced mathematicians don't have problems with the meaning of 'series'. ".
Yes, they can distinguish between a convergent sequence and a convergent series. They know that 'sum of a sequence' means (about?) the same as 'sum of a series'. And they know that you should never say 'absolutely convergent sequence'. Nor 'limit of a series'. But do they all know the difference between a Cesàro-convergent sequence and a Cesàro-convergent series ? See the last sentence before the heading "Examples" in
[1]: "For any convergent sequence, the corresponding series is Cesàro summable and the limit of the sequence coincides with the Cesàro sum."
D. Is it a mathematical problem or a pedagogical problem that caused the situation that writers and controllers of the English WP-article on 'series', gave up their attempts to formulate a definition? I don't know.
E. On the meaning of 'summable sequence': Bourbaki choose for summable = absolutely summable ! For evidence, scroll down to the last five lines of page 269 in this book. (The same in the first (French) edition, 1942.)
F. About writing an article on this subject: ten years ago I had
this article in the journal "Nieuw Archief voor wiskunde" of the Dutch Mathematical Society (title: Was Reihen sind, kann man nicht sagen; reeks = series, rij = sequence)
--
Hesselp (
talk) 20:43, 29 September 2017 (UTC)
On your remarks (5 October 2017) until "Identify ..." : No problem for me to agree with you that a mathematical notion can be defined in differntly formulated - equivalent - ways.
On your sentence: "Identify a series with the sequence of its terms, or the sequence of its partial sums, or with the pair of sequences, it is all the same, due to the evident one-to-one correspondence between these objects." I read this as:
A series can be identified/represented by its terms,
a series can be identified/represented by its partial sums, and
a series can be identified/represented by its terms-sums-couple.
But the mathematical notion mostly called 'series' isn't defined by showing its terms representation. Nor by showing - imo - its partial-sums representation or its terms-and-sums representation.
One more point: I hope I don't misread your sentence by interpreting 'these objects' as: 'these representations' or 'these identification tools'.
Substituting the word 'series' in your sentence by the word 'sequence', the content of it remains equally true - yes? This seems to indicate that you see the notion series as identical with the notion sequence - yes?
If your answer is 'yes', I agree with 'Only a pedagogical problem' (without question mark). I.e.: the problem of how to convince quite a lot of authors of calculus-books (and WP-articles) that their way of presenting this notion-with-two-names in their separate chapters 'Sequences' and 'Series' makes it very difficult for students to see the definitions in the two chapters as intended to be equivalent. --
Hesselp (
talk) 21:56, 8 October 2017 (UTC)
Welcome to Wikipedia. Everyone is welcome to contribute constructively to the encyclopedia. However, talk pages are meant to be a record of a discussion; deleting or editing legitimate comments, as you did at User talk:Kudpung, is considered bad practice, even if you meant well. Even making spelling and grammatical corrections in others' comments is generally frowned upon, as it tends to irritate the users whose comments you are correcting. Take a look at the welcome page to learn more about contributing to this encyclopedia. Thank you. — O Fortuna semper crescis, aut decrescis 15:18, 23 June 2017 (UTC)
Hesselp, please stop your incompetent edits to series related articles. I recently had to revert your edit to Cesaro summation, since you obviously have no idea what that is. Sławomir Biały ( talk) 00:26, 10 October 2017 (UTC)
There is currently a discussion at Wikipedia:Administrators' noticeboard/Incidents regarding an issue with which you may have been involved. D.Lazard ( talk) 16:57, 30 October 2017 (UTC)
I've closed your ANI thread with consensus to indefinitely topic ban you from all articles on or related to mathematical series. Effective immediately, if you are found to be editing any mathematical series article, its talk page, or anything else remotely related to mathematical series you will be blocked. A record of your ban has been noted at Wikipedia:Editing restrictions/Placed by the Wikipedia community, and you will remained topic banned until the community elects to let you edit the articles again or until the arbitration committee takes up the case. TomStar81 ( Talk) 22:53, 7 November 2017 (UTC)
There is currently a discussion at Wikipedia:Administrators' noticeboard/Incidents regarding an issue with which you may have been involved. Sławomir Biały ( talk) 23:09, 22 November 2017 (UTC)
Please stop trying to add material at e (mathematical constant). Consensus was clearly against the edits you wanted to make back in November when you stopped editing. I thought you understood then, but it would seem you don't. – Deacon Vorbis ( carbon • videos) 15:57, 27 April 2018 (UTC)
Your recent editing history shows that you are currently engaged in an edit war. To resolve the content dispute, please do not revert or change the edits of others when you are reverted. Instead of reverting, please use the talk page to work toward making a version that represents consensus among editors. The best practice at this stage is to discuss, not edit-war. See BRD for how this is done. If discussions reach an impasse, you can then post a request for help at a relevant noticeboard or seek dispute resolution. In some cases, you may wish to request temporary page protection.
Being involved in an edit war can result in your being blocked from editing—especially if you violate the three-revert rule, which states that an editor must not perform more than three reverts on a single page within a 24-hour period. Undoing another editor's work—whether in whole or in part, whether involving the same or different material each time—counts as a revert. Also keep in mind that while violating the three-revert rule often leads to a block, you can still be blocked for edit warring—even if you don't violate the three-revert rule—should your behavior indicate that you intend to continue reverting repeatedly. MrOllie ( talk) 21:37, 28 April 2017 (UTC)
There is currently a discussion at Wikipedia:Administrators' noticeboard/Incidents regarding an issue with which you may have been involved. Sławomir Biały ( talk) 11:58, 6 May 2017 (UTC)
Oops... you'd better not violate 3RR. Boris Tsirelson ( talk) 20:20, 12 May 2017 (UTC)
Hi. I'm just informing you that following the ANI discussion, the consensus by the community is that you be Topic baned for a period of six (6) months starting today, from editing the article at Series (mathematics) and its talk page. Any infringement of this ban will result in your account being blocked without warning for a duration of the administrator's discretion. This sanction is imposed in my capacity as an uninvolved administrator. You may appeal this sanction at the administrators' noticeboard. You may also appeal directly to me (on my talk page), before or instead of appealing to the noticeboard. The ban remains in force however, until any official change is pronounced. Kudpung กุดผึ้ง ( talk) 10:35, 28 May 2017 (UTC)
Hi, Hessel Pot. Mathematically, I like the point of view pushed by you. I happen to teach analysis only in the second year. But if I taught it in the first year, maybe I would follow you, saying "summable sequence" and "series representation", never just "series".
However, note a difference: on my courses I am the decision maker; here on Wikipedia I am not. Here a point of view cannot be presented until/unless it is widely used. And if it is, it must be presented with "due weight". Boris Tsirelson ( talk) 18:52, 28 May 2017 (UTC)
(Unindent) To
Boris Tsirelson. About the past and the present:
On the Dutch WP the dominant opinion is, that by 'the harmonic series' another mathematical concept is meant than by 'the harmonic sequence'; that the two names are not interchangeable. Why? Because the way the are notated is different (terms separated by comma's vs. pluses). My contra-argument that - at least up to my school-years - the word 'sequence' ('rij' in Dutch) was almost non-existant, is rejected as not relevant now. (About Fibonacci sequence versus Fibonacci series we didn't discuss.)
I added 'calculatable' (a spontaneous invention, as a variant on your 'writable') after I saw Swokowski's sentence (p. 533): "To calculate S5, S6, S7, and so on, we add more terms of the series.". With the word 'calculate' he suggest an action/activity (comparable with the 'addition-action' in primary school), which is not possible to perform in the general case. Imo Sw. leads the reader into a wrong direction with this 'calculate'.
The same wrong direction is suggested (imo) by the use of 'infinite sum' in the key-sentence of the series-article. The existence of a mathematical concept named 'infinite sum' is suggested, being the result of some kind of calculating/working with the given terms.
I did not at all mean 'countable number'. --
Hesselp (
talk) 23:18, 26 June 2017 (UTC)
- I. Are you aware of the fact (is it a fact?) that the word 'series' is very often used in direct connection with (mostly directly followed by) a symbolic, written form of type 'capital-sigma' or type 'pluses-bullets'
( Σn=1∞ or or variants).
- II. As a consequence of I, the word 'series' will be absent in the oral/verbal part of your lectures on calculus (without a blackboard or a more modern pictural divice for communication). I don't have a verbal alternative for a capital-sigma-form or a pluses-bullets-form. --
Hesselp (
talk) 21:43, 28 June 2017 (UTC)
- III. Can you imagine that I see the facts described in I and II as a strong indication that the word 'series' is not in use for a mathematical concept, but for a notational form?
- IV. With your three definition-variants (1, 1', and 1") you want to illustrate that, in your view/interpretation, the terms-sequence (an) and the sums-sequence (Sn) have an equivalent role in the concept/object 'series'. Right? I paraphrase your triple by:
"Definition: An ordered pair of real-number-sequences an; bn , related by bn = a1 + ··· + an (or the equivalent: a1 = b1, an+1 = bn +1 − bn ), is called a (real-number-)series." Right?
- V. Why this emphasis on the equivalent role of both sequences? The relation between a (summable) terms-sequence and the limit of its sums-sequence is an important one in calculus. As is the relation between a (complicated) function and the sequence of its (less complicated) Maclaurin-terms. As well as the relation between f and the sequence of the Fourier-terms of f. On the contrary, the relation between the sums-sequence and the limit of its terms-sequence has no applications in calculus (as far as I know). So I don't see a good reason to emphasize an equivalency of terms-sequence and sums-sequence. Right?
- VI. I've another argument against adopting the pair of sequences as being the heart of a 'series'-concept. Isn't it standard use in mathematics to formulate definitions as simple/elementary as possible? So why should the sums-sequence be mentioned in the definition, while this sums-sequence is already completely determined by the other sequence?
- VII. Your pair-of-sequences as being the mathematical meaning of 'series', I met earlier in calculus-books by: Creighton Buck 1956, Zamansky '58, Apostol '74, Maurin '76, Protter/Morrey '77, Encyclopaedia of Mathematics '92, Gaughan '98, Boos '00, Edward Azoff '05. Not long ago I red that the source is in a Bourbaki-publication. (I cannot find back where I saw this; do you know this source?) As I said in VI, this definition doesn't say anything more than 'series' is another word for 'sequence'. Or more precise: A sequence with a sums-sequence (the same as: a sequence with additionable terms) is called series.
Or can you describe a difference between "a terms-sequence sums-sequence pair" and "a sequence with terms that can be added to form its sums-sequence" (essentially the same as Cauchy's: "une suite de nombres réels") ? --
Hesselp (
talk) 21:43, 28 June 2017 (UTC)
- VIII. Once more on your: "How do I interpret the word "series" when reading or writing an article in a mathematical journal?".
Can you find back places in articles/books written by you (or red by you) where you used/red the word 'series' ? (The articles/books not being tutorial texts on calculus.) Can at that places the word 'series' be red as: "the combination of expressions for the summation function and for a sequence"? Or can "series Σ a " (or "series a1 + a2 + ··· ") be red as "sequence a" ? --
Hesselp (
talk) 05:36, 29 June 2017 (UTC)
(I wrote my 'point VIII' before your '(Unindent)' came in, but was just some minutes late in posting it.)
- IX. On your: "It seems we agreed to put metamathematical notions aside (wherever possible).": I see this put aside as not possible in a discussion on the way mathematicians use the word 'series' in a calculus context. The consequence of this put aside should be (imo) that you want to skip WP-articles with titles: 'sum', 'product', 'quotient', 'logarithm', 'integral', 'root', 'power', et cetera.
- X. On your: "...surely I could pronounce..." and your "at some point I would say [!,HP]...": Do you really say (verbally, in words), with your back to the blackboard: "series Greek capital letter sigma indexed by n is one and by infinite a indexed n " ? Or something like this, starting with the word 'series' ? --
Hesselp (
talk) 07:03, 29 June 2017 (UTC)
Sorry, it is impractical to discuss 8 points in parallel. Let us close one point and then start another.
Boris Tsirelson (
talk) 06:27, 29 June 2017 (UTC)
Yes, it can be practical to discuss points I - X (maybe some combined) in different (sub)sections. --
Hesselp (
talk) 07:03, 29 June 2017 (UTC)
Well, then create the subsections; but anyway I do not want to scatter, and will attend one subsection at a time. The more so that the conclusion reached in one subsection may affect the discussion of another. Thus, choose their order...
Boris Tsirelson (
talk) 07:11, 29 June 2017 (UTC)</math>
I leave it to you, Tsirel, to choose the order. Or else: why not the order of the roman numbers? --
Hesselp (
talk) 08:09, 29 June 2017 (UTC)
OK, I start the split process.
Boris Tsirelson (
talk) 09:14, 29 June 2017 (UTC)
Maybe I understand your point, at last!
Here is my guess. We have here another oddity of mathematical terminology. (Strangely I did not note it before.) First, both notations, and are ambiguous; depending on the context they may denote either a series (that has terms and partial sums), or the sum of this series! Examples:
in (a) the expression denotes the series, but in (b) a similar expression denotes the sum of the series.
Is the word "series" ambiguous in the same way? Probably, less ambiguous, since usually one writes "sum of the series", not just "series", when the sum is meant. But probably a less accurate language is used sometimes, like this:
in this phrase the "series" does not exceed a number, therefore the "series" is interpreted as a number. Though, one may say that no, the author uses element-wise inequality between one series and another series, and deduces inequality between their sums.
Does a similar oddity apply to integrals? Usually, denotes the number (rather than the whole integrand function). But probably a less accurate language is used sometimes, like this:
surely in (d) one does not mean Riemann sums for the number and in (e) one does not mean that a number could be absolutely convergent. Boris Tsirelson ( talk) 18:59, 29 June 2017 (UTC)
@
Tsirel
A. "it is not about contradictions in mathematics [...] but about oddities of mathematical terminology"
A mathematical or a pedagogical problem? I cannot choose. But in the present text of
de:Reihe (Mathematik) I read in Satz 3 that 'Reihe' is the name for every Folge with a Differenzenfolge (e.g. every Zahlenfolge). And in Satz 8 (in a cripple way) that 'Reihe zum Folge (an)' means the same as 'Partialsummenfolge der Folge (an)' . Sätz 3 and 8 are contradictional, imo. At least it should be noticed that this are two different ways the word 'Reihe' is used in calculus.
B. How do first class mathematicians (try to) solve this definition-question? See:
- Bourbaki, Éléments de Mathématique, Première partie, Livre III (Topologie générale), Chap. 3, Par. 4, No 6. Series (Deuxième Édition, 1951, p. 42-43) "On appelle série définie par la suite (xn) le couple des suites (xn) et (sn) ainsi associées."
-
Encyclopedia of Mathematics - Series: "A pair of sequences of complex numbers {an} and {a1+ ··· + an} is called a (simple) series of numbers"
- B. Tsirelson
28 June 2017: "Definition 1. A series (of real numbers) is a pair of (infinite) sequences and (of real numbers) such that for each ."
This definition - though formally not incorrect - is absurd, for you can choose whatever you want as second element in the pair. "Series" can be defined as being an ordered pair with an infinite sequence-with-addition as its first element and a arbitrary object as its second. Consequences:
an alternating/harmonic/Fibonacci series a pair with an alternating/harmonic/Fibonacci sequence as its first element;
a convergent series a pair with as first element a sequence with converging partial sums;
the terms / partial sums of a series the terms / partial sums of its first element
the sum of a series ( (an) ; .... ) the limit of the partial sums of sequence (an).
A series is a pair? Did you ever see (1, 1/2, 1/4, 1/8, ... ; 1, 3/2, 7/4, 15/8, ...) as an example of a convergent series?
C. "Experienced mathematicians don't have problems with the meaning of 'series'. ".
Yes, they can distinguish between a convergent sequence and a convergent series. They know that 'sum of a sequence' means (about?) the same as 'sum of a series'. And they know that you should never say 'absolutely convergent sequence'. Nor 'limit of a series'. But do they all know the difference between a Cesàro-convergent sequence and a Cesàro-convergent series ? See the last sentence before the heading "Examples" in
[1]: "For any convergent sequence, the corresponding series is Cesàro summable and the limit of the sequence coincides with the Cesàro sum."
D. Is it a mathematical problem or a pedagogical problem that caused the situation that writers and controllers of the English WP-article on 'series', gave up their attempts to formulate a definition? I don't know.
E. On the meaning of 'summable sequence': Bourbaki choose for summable = absolutely summable ! For evidence, scroll down to the last five lines of page 269 in this book. (The same in the first (French) edition, 1942.)
F. About writing an article on this subject: ten years ago I had
this article in the journal "Nieuw Archief voor wiskunde" of the Dutch Mathematical Society (title: Was Reihen sind, kann man nicht sagen; reeks = series, rij = sequence)
--
Hesselp (
talk) 20:43, 29 September 2017 (UTC)
On your remarks (5 October 2017) until "Identify ..." : No problem for me to agree with you that a mathematical notion can be defined in differntly formulated - equivalent - ways.
On your sentence: "Identify a series with the sequence of its terms, or the sequence of its partial sums, or with the pair of sequences, it is all the same, due to the evident one-to-one correspondence between these objects." I read this as:
A series can be identified/represented by its terms,
a series can be identified/represented by its partial sums, and
a series can be identified/represented by its terms-sums-couple.
But the mathematical notion mostly called 'series' isn't defined by showing its terms representation. Nor by showing - imo - its partial-sums representation or its terms-and-sums representation.
One more point: I hope I don't misread your sentence by interpreting 'these objects' as: 'these representations' or 'these identification tools'.
Substituting the word 'series' in your sentence by the word 'sequence', the content of it remains equally true - yes? This seems to indicate that you see the notion series as identical with the notion sequence - yes?
If your answer is 'yes', I agree with 'Only a pedagogical problem' (without question mark). I.e.: the problem of how to convince quite a lot of authors of calculus-books (and WP-articles) that their way of presenting this notion-with-two-names in their separate chapters 'Sequences' and 'Series' makes it very difficult for students to see the definitions in the two chapters as intended to be equivalent. --
Hesselp (
talk) 21:56, 8 October 2017 (UTC)
Welcome to Wikipedia. Everyone is welcome to contribute constructively to the encyclopedia. However, talk pages are meant to be a record of a discussion; deleting or editing legitimate comments, as you did at User talk:Kudpung, is considered bad practice, even if you meant well. Even making spelling and grammatical corrections in others' comments is generally frowned upon, as it tends to irritate the users whose comments you are correcting. Take a look at the welcome page to learn more about contributing to this encyclopedia. Thank you. — O Fortuna semper crescis, aut decrescis 15:18, 23 June 2017 (UTC)
Hesselp, please stop your incompetent edits to series related articles. I recently had to revert your edit to Cesaro summation, since you obviously have no idea what that is. Sławomir Biały ( talk) 00:26, 10 October 2017 (UTC)
There is currently a discussion at Wikipedia:Administrators' noticeboard/Incidents regarding an issue with which you may have been involved. D.Lazard ( talk) 16:57, 30 October 2017 (UTC)
I've closed your ANI thread with consensus to indefinitely topic ban you from all articles on or related to mathematical series. Effective immediately, if you are found to be editing any mathematical series article, its talk page, or anything else remotely related to mathematical series you will be blocked. A record of your ban has been noted at Wikipedia:Editing restrictions/Placed by the Wikipedia community, and you will remained topic banned until the community elects to let you edit the articles again or until the arbitration committee takes up the case. TomStar81 ( Talk) 22:53, 7 November 2017 (UTC)
There is currently a discussion at Wikipedia:Administrators' noticeboard/Incidents regarding an issue with which you may have been involved. Sławomir Biały ( talk) 23:09, 22 November 2017 (UTC)
Please stop trying to add material at e (mathematical constant). Consensus was clearly against the edits you wanted to make back in November when you stopped editing. I thought you understood then, but it would seem you don't. – Deacon Vorbis ( carbon • videos) 15:57, 27 April 2018 (UTC)