Collection detailing various sums of reciprocal numbers in mathematics
In
mathematics and especially
number theory, the sum of reciprocals generally is computed for the
reciprocals of some or all of the
positiveintegers (counting numbers)—that is, it is generally the sum of
unit fractions. If infinitely many numbers have their reciprocals summed, generally the terms are given in a certain sequence and the first n of them are summed, then one more is included to give the sum of the first n+1 of them, etc.
If only finitely many numbers are included, the key issue is usually to find a simple expression for the value of the sum, or to require the sum to be less than a certain value, or to determine whether the sum is ever an integer.
For an
infinite series of reciprocals, the issues are twofold: First, does the sequence of sums
diverge—that is, does it eventually exceed any given number—or does it
converge, meaning there is some number that it gets arbitrarily close to without ever exceeding it? (A set of positive integers is said to be
large if the sum of its reciprocals diverges, and small if it converges.) Second, if it converges, what is a simple expression for the value it converges to, is that value
rational or
irrational, and is that value
algebraic or
transcendental?[1]
Finitely many terms
The
harmonic mean of a set of positive integers is the number of numbers times the reciprocal of the sum of their reciprocals.
The
optic equation requires the sum of the reciprocals of two positive integers a and b to equal the reciprocal of a third positive integer c. All solutions are given by a = mn + m2, b = mn + n2, c = mn. This equation appears in various contexts in elementary
geometry.
The
Fermat–Catalan conjecture concerns a certain
Diophantine equation, equating the sum of two terms, each a positive integer raised to a positive integer power, to a third term that is also a positive integer raised to a positive integer power (with the base integers having no prime factor in common). The conjecture asks whether the equation has an infinitude of solutions in which the sum of the reciprocals of the three exponents in the equation must be less than 1. The purpose of this restriction is to preclude the known infinitude of solutions in which two exponents are 2 and the other exponent is any even number.
The n-th
harmonic number, which is the sum of the reciprocals of the first n positive integers, is never an integer except for the case n = 1.
Moreover,
József Kürschák proved in 1918 that the sum of the reciprocals of consecutive natural numbers (whether starting from 1 or not) is never an integer.
There are
14 distinct combinations of four integers such that the sum of their reciprocals is 1, of which six use four distinct integers and eight repeat at least one integer.
An
Egyptian fraction is the sum of a finite number of reciprocals of positive integers. According to the proof of the
Erdős–Graham problem, if the set of
integers greater than one is
partitioned into finitely many subsets, then one of the subsets can be used to form an
Egyptian fraction representation of 1.
The
Erdős–Straus conjecture states that for all integers n ≥ 2, the rational number 4/n can be expressed as the sum of three reciprocals of positive integers.
The
Fermat quotient with base 2, which is for odd prime p, when expressed in
modp and multiplied by –2, equals the sum of the reciprocals mod p of the numbers lying in the first half of the range {1, p − 1}.
In any
triangle, the sum of the reciprocals of the
altitudes equals the reciprocal of the
radius of the
incircle (regardless of whether or not they are integers).
In a
right triangle, the sum of the reciprocals of the squares of the altitudes from the legs (equivalently, of the squares of the legs themselves) equals the reciprocal of the square of the altitude from the hypotenuse (the
inverse Pythagorean theorem). This holds whether or not the numbers are integers; there is a formula (see
here) that generates all integer cases.
A triangle not necessarily in the
Euclidean plane can be specified as having angles and Then the triangle is in Euclidean space if the sum of the reciprocals of p, q, and r equals 1,
spherical space if that sum is greater than 1, and
hyperbolic space if the sum is less than 1.
A
harmonic divisor number is a positive integer whose divisors have a
harmonic mean that is an integer. The first five of these are 1, 6, 28, 140, and 270. It is not known whether any harmonic divisor numbers (besides 1) are odd, but there are no odd ones less than 1024.
When eight points are distributed on the surface of a
sphere with the aim of maximizing the distance between them in some sense, the resulting shape corresponds to a
square antiprism. Specific methods of distributing the points include, for example, minimizing the sum of all reciprocals of squares of distances between points.
A
sum-free sequence of increasing positive integers is one for which no number is the sum of any
subset of the previous ones. The sum of the reciprocals of the numbers in any sum-free sequence is less than 2.8570 .
The sum of the reciprocals of the
twin primes, of which there may be finitely many or infinitely many, is known to be finite and is called
Brun's constant, approximately 1.9022 . The reciprocal of five conventionally appears twice in the sum.
The sum of the reciprocals of the
Proth primes, of which there may be finitely many or infinitely many, is known to be finite, approximately 0.747392479 .[2]
The
prime quadruplets are pairs of twin primes with only one odd number between them. The sum of the reciprocals of the numbers in prime quadruplets is approximately 0.8706 .
The sum of the reciprocals of the
perfect powers (including duplicates) is 1 .
The sum of the reciprocals of the
perfect powers (excluding duplicates) is approximately 0.8745 .[3]
The sum of the reciprocals of the powers is approximately equal to 1.2913 . The sum is exactly equal to a definite
integral:
The
Goldbach–Euler theorem states that the sum of the reciprocals of the numbers that are 1 less than a perfect power (excluding duplicates) is 1 .
The sum of the reciprocals of all the non-zero
triangular numbers is 2 .
The
reciprocal Fibonacci constant is the sum of the reciprocals of the
Fibonacci numbers, which is known to be finite and irrational and approximately equal to 3.3599 . For other finite sums of subsets of the reciprocals of Fibonacci numbers, see
here.
An
exponential factorial is an operation
recursively defined as For example, where the exponents are evaluated from the top down. The sum of the reciprocals of the exponential factorials from 1 onward is approximately 1.6111 and is transcendental.
A "
powerful number" is a positive integer for which every prime appearing in its
prime factorization appears there at least twice. The sum of the reciprocals of the powerful numbers is close to 1.9436 .[4]
The sum of the reciprocals of the cubes of positive integers is called
Apéry's constantζ(3) , and equals approximately 1.2021 . This number is
irrational, but it is not known whether or not it is
transcendental.
The reciprocals of the non-negative integer
powers of 2 sum to 2 . This is a particular case of the sum of the reciprocals of any geometric series where the first term and the common ratio are positive integers. If the first term is a and the common ratio is r then the sum is r/ a (r − 1) .
The
Kempner series is the sum of the reciprocals of all positive integers not containing the digit "9" in base 10 . Unlike the
harmonic series, which does not exclude those numbers, this series converges, specifically to approximately 22.9207 .
A
palindromic number is one that remains the same when its digits are reversed. The sum of the reciprocals of the palindromic numbers converges to approximately 3.3703 .
A
pentatope number is a number in the fifth cell of any row of
Pascal's triangle starting with the five-term row 1 4 6 4 1 . The sum of the reciprocals of the pentatope numbers is 4/ 3 .
Sylvester's sequence is an
integer sequence in which each member of the sequence is the product of the previous members, plus one. The first few terms of the sequence are 2, 3, 7, 43, 1807 . The sum of the reciprocals of the numbers in Sylvester's sequence is 1 .
to an analytic function on the entire complex plane except for s = 1, where ζ(s) has a pole. This series converges if and only if the real part of s is greater than 1 .
The sum of the reciprocals of the
pronic numbers (products of two consecutive integers) (excluding 0) is 1 (see
Telescoping series).
Divergent series
The n-th partial sum of the
harmonic series, which is the sum of the reciprocals of the first n positive integers, diverges as n goes to infinity, albeit extremely slowly: The sum of the first 1043 terms is less than 100 . The difference between the cumulative sum and the
natural logarithm of n converges to the
Euler–Mascheroni constant, commonly denoted as which is approximately 0.5772 .
Similarly, the sum of the reciprocals of the primes of the form 4n + 1 is divergent. By
Fermat's theorem on sums of two squares, it follows that the sum of reciprocals of numbers of the form where a and b are non-negative integers, not both equal to 0, diverges, with or without repetition.
If a(k) is any ascending series of positive integers with the property that there exists N such that a(k + 1) − a(k) < N for all k then the sum of the reciprocals 1/a(k) diverges.
^Unless given here, references are in the linked articles.
^Borsos, Bertalan; Kovács, Attila; Tihanyi, Norbert (1 September 2022). "Tight upper and lower bounds for the reciprocal sum of Proth primes". The Ramanujan Journal. 59 (1): 181–198.
doi:
10.1007/s11139-021-00536-2.
hdl:10831/83020.
S2CID246024152.
Collection detailing various sums of reciprocal numbers in mathematics
In
mathematics and especially
number theory, the sum of reciprocals generally is computed for the
reciprocals of some or all of the
positiveintegers (counting numbers)—that is, it is generally the sum of
unit fractions. If infinitely many numbers have their reciprocals summed, generally the terms are given in a certain sequence and the first n of them are summed, then one more is included to give the sum of the first n+1 of them, etc.
If only finitely many numbers are included, the key issue is usually to find a simple expression for the value of the sum, or to require the sum to be less than a certain value, or to determine whether the sum is ever an integer.
For an
infinite series of reciprocals, the issues are twofold: First, does the sequence of sums
diverge—that is, does it eventually exceed any given number—or does it
converge, meaning there is some number that it gets arbitrarily close to without ever exceeding it? (A set of positive integers is said to be
large if the sum of its reciprocals diverges, and small if it converges.) Second, if it converges, what is a simple expression for the value it converges to, is that value
rational or
irrational, and is that value
algebraic or
transcendental?[1]
Finitely many terms
The
harmonic mean of a set of positive integers is the number of numbers times the reciprocal of the sum of their reciprocals.
The
optic equation requires the sum of the reciprocals of two positive integers a and b to equal the reciprocal of a third positive integer c. All solutions are given by a = mn + m2, b = mn + n2, c = mn. This equation appears in various contexts in elementary
geometry.
The
Fermat–Catalan conjecture concerns a certain
Diophantine equation, equating the sum of two terms, each a positive integer raised to a positive integer power, to a third term that is also a positive integer raised to a positive integer power (with the base integers having no prime factor in common). The conjecture asks whether the equation has an infinitude of solutions in which the sum of the reciprocals of the three exponents in the equation must be less than 1. The purpose of this restriction is to preclude the known infinitude of solutions in which two exponents are 2 and the other exponent is any even number.
The n-th
harmonic number, which is the sum of the reciprocals of the first n positive integers, is never an integer except for the case n = 1.
Moreover,
József Kürschák proved in 1918 that the sum of the reciprocals of consecutive natural numbers (whether starting from 1 or not) is never an integer.
There are
14 distinct combinations of four integers such that the sum of their reciprocals is 1, of which six use four distinct integers and eight repeat at least one integer.
An
Egyptian fraction is the sum of a finite number of reciprocals of positive integers. According to the proof of the
Erdős–Graham problem, if the set of
integers greater than one is
partitioned into finitely many subsets, then one of the subsets can be used to form an
Egyptian fraction representation of 1.
The
Erdős–Straus conjecture states that for all integers n ≥ 2, the rational number 4/n can be expressed as the sum of three reciprocals of positive integers.
The
Fermat quotient with base 2, which is for odd prime p, when expressed in
modp and multiplied by –2, equals the sum of the reciprocals mod p of the numbers lying in the first half of the range {1, p − 1}.
In any
triangle, the sum of the reciprocals of the
altitudes equals the reciprocal of the
radius of the
incircle (regardless of whether or not they are integers).
In a
right triangle, the sum of the reciprocals of the squares of the altitudes from the legs (equivalently, of the squares of the legs themselves) equals the reciprocal of the square of the altitude from the hypotenuse (the
inverse Pythagorean theorem). This holds whether or not the numbers are integers; there is a formula (see
here) that generates all integer cases.
A triangle not necessarily in the
Euclidean plane can be specified as having angles and Then the triangle is in Euclidean space if the sum of the reciprocals of p, q, and r equals 1,
spherical space if that sum is greater than 1, and
hyperbolic space if the sum is less than 1.
A
harmonic divisor number is a positive integer whose divisors have a
harmonic mean that is an integer. The first five of these are 1, 6, 28, 140, and 270. It is not known whether any harmonic divisor numbers (besides 1) are odd, but there are no odd ones less than 1024.
When eight points are distributed on the surface of a
sphere with the aim of maximizing the distance between them in some sense, the resulting shape corresponds to a
square antiprism. Specific methods of distributing the points include, for example, minimizing the sum of all reciprocals of squares of distances between points.
A
sum-free sequence of increasing positive integers is one for which no number is the sum of any
subset of the previous ones. The sum of the reciprocals of the numbers in any sum-free sequence is less than 2.8570 .
The sum of the reciprocals of the
twin primes, of which there may be finitely many or infinitely many, is known to be finite and is called
Brun's constant, approximately 1.9022 . The reciprocal of five conventionally appears twice in the sum.
The sum of the reciprocals of the
Proth primes, of which there may be finitely many or infinitely many, is known to be finite, approximately 0.747392479 .[2]
The
prime quadruplets are pairs of twin primes with only one odd number between them. The sum of the reciprocals of the numbers in prime quadruplets is approximately 0.8706 .
The sum of the reciprocals of the
perfect powers (including duplicates) is 1 .
The sum of the reciprocals of the
perfect powers (excluding duplicates) is approximately 0.8745 .[3]
The sum of the reciprocals of the powers is approximately equal to 1.2913 . The sum is exactly equal to a definite
integral:
The
Goldbach–Euler theorem states that the sum of the reciprocals of the numbers that are 1 less than a perfect power (excluding duplicates) is 1 .
The sum of the reciprocals of all the non-zero
triangular numbers is 2 .
The
reciprocal Fibonacci constant is the sum of the reciprocals of the
Fibonacci numbers, which is known to be finite and irrational and approximately equal to 3.3599 . For other finite sums of subsets of the reciprocals of Fibonacci numbers, see
here.
An
exponential factorial is an operation
recursively defined as For example, where the exponents are evaluated from the top down. The sum of the reciprocals of the exponential factorials from 1 onward is approximately 1.6111 and is transcendental.
A "
powerful number" is a positive integer for which every prime appearing in its
prime factorization appears there at least twice. The sum of the reciprocals of the powerful numbers is close to 1.9436 .[4]
The sum of the reciprocals of the cubes of positive integers is called
Apéry's constantζ(3) , and equals approximately 1.2021 . This number is
irrational, but it is not known whether or not it is
transcendental.
The reciprocals of the non-negative integer
powers of 2 sum to 2 . This is a particular case of the sum of the reciprocals of any geometric series where the first term and the common ratio are positive integers. If the first term is a and the common ratio is r then the sum is r/ a (r − 1) .
The
Kempner series is the sum of the reciprocals of all positive integers not containing the digit "9" in base 10 . Unlike the
harmonic series, which does not exclude those numbers, this series converges, specifically to approximately 22.9207 .
A
palindromic number is one that remains the same when its digits are reversed. The sum of the reciprocals of the palindromic numbers converges to approximately 3.3703 .
A
pentatope number is a number in the fifth cell of any row of
Pascal's triangle starting with the five-term row 1 4 6 4 1 . The sum of the reciprocals of the pentatope numbers is 4/ 3 .
Sylvester's sequence is an
integer sequence in which each member of the sequence is the product of the previous members, plus one. The first few terms of the sequence are 2, 3, 7, 43, 1807 . The sum of the reciprocals of the numbers in Sylvester's sequence is 1 .
to an analytic function on the entire complex plane except for s = 1, where ζ(s) has a pole. This series converges if and only if the real part of s is greater than 1 .
The sum of the reciprocals of the
pronic numbers (products of two consecutive integers) (excluding 0) is 1 (see
Telescoping series).
Divergent series
The n-th partial sum of the
harmonic series, which is the sum of the reciprocals of the first n positive integers, diverges as n goes to infinity, albeit extremely slowly: The sum of the first 1043 terms is less than 100 . The difference between the cumulative sum and the
natural logarithm of n converges to the
Euler–Mascheroni constant, commonly denoted as which is approximately 0.5772 .
Similarly, the sum of the reciprocals of the primes of the form 4n + 1 is divergent. By
Fermat's theorem on sums of two squares, it follows that the sum of reciprocals of numbers of the form where a and b are non-negative integers, not both equal to 0, diverges, with or without repetition.
If a(k) is any ascending series of positive integers with the property that there exists N such that a(k + 1) − a(k) < N for all k then the sum of the reciprocals 1/a(k) diverges.
^Unless given here, references are in the linked articles.
^Borsos, Bertalan; Kovács, Attila; Tihanyi, Norbert (1 September 2022). "Tight upper and lower bounds for the reciprocal sum of Proth primes". The Ramanujan Journal. 59 (1): 181–198.
doi:
10.1007/s11139-021-00536-2.
hdl:10831/83020.
S2CID246024152.