View source for List of sums of reciprocals

{{Short description|Collection detailing various sums of reciprocal numbers in mathematics}}
In [[Mathematics|mathematics]] and especially [[Number theory|number theory]], the '''sum of reciprocals''' generally is computed for the [[Multiplicative inverse|reciprocal]]s of some or all of the positive [[Integer|integer]]s (counting numbers)&mdash;that is, it is generally the sum of [[Unit fraction|unit fraction]]s.  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 [[Divergent series|diverge]]—that is, does it eventually exceed any given number—or does it [[Convergent series|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 set (combinatorics)|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 number|rational]] or [[Irrational number|irrational]], and is that value [[Algebraic number|algebraic]] or [[Transcendental number|transcendental]]?<ref>Unless given here, references are in the linked articles.</ref>

==Finitely many terms==

*The [[Harmonic mean|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|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''&nbsp;=&nbsp;''mn''&nbsp;+&nbsp;''m''<sup>2</sup>, ''b''&nbsp;=&nbsp;''mn''&nbsp;+&nbsp;''n''<sup>2</sup>, ''c''&nbsp;=&nbsp;''mn''. This equation appears in various contexts in elementary [[Geometry|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|harmonic number]], which is the sum of the reciprocals of the first ''n'' positive integers, is never an integer except for the case&nbsp;''n''&nbsp;=&nbsp;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.
*The sum of the reciprocals of the first ''n'' primes is not an integer for any ''n''.
*There are [[215 (number)|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 [[Integer|integer]]s greater than one is [[Partition of a set|partition]]ed 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''&nbsp;≥&nbsp;2, the rational number 4/''n''  can be expressed as the sum of three reciprocals of positive integers.
*The [[Fermat quotient#Special values|Fermat quotient]] with base 2, which is <math>\frac{2^{p-1}-1}{p}</math> for odd prime ''p'', when expressed in [[Modular arithmetic|mod]] ''p'' and multiplied by –2, equals the sum of the reciprocals mod&nbsp;''p'' of the numbers lying in the first half of the range&nbsp;{1,&nbsp;''p''&nbsp;&minus;&nbsp;1}.
*In any [[Triangle|triangle]], the sum of the reciprocals of the [[Altitude (triangle)#Inradius theorems|altitudes]] equals the reciprocal of the [[Radius|radius]] of the incircle (regardless of whether or not they are integers).
*In a [[Right triangle|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|inverse Pythagorean theorem]]). This holds whether or not the numbers are integers; there is a formula (see [[Integer triangle#Pythagorean triangles with integer altitude from the hypotenuse|here]]) that generates all integer cases.
*A triangle not necessarily in the [[Euclidean plane]] can be specified as having angles <math>\frac{\pi}{p},</math> <math>\frac{\pi}{q},</math> and <math>\frac{\pi}{r}.</math> Then the triangle is in Euclidean space if the sum of the reciprocals of ''p, q,'' and ''r'' equals 1, [[Spherical trigonometry|spherical space]] if that sum is greater than 1, and [[Hyperbolic geometry|hyperbolic space]] if the sum is less than 1.
*A [[Harmonic divisor number|harmonic divisor number]] is a positive integer whose divisors have a [[Harmonic mean|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 10<sup>24</sup>.
* The sum of the reciprocals of the [[Divisor|divisor]]s of a [[Perfect number|perfect number]] is 2.
*When eight points are distributed on the surface of a [[Sphere|sphere]] with the aim of maximizing the distance between them in some sense, the resulting shape corresponds to a [[Square antiprism|square antiprism]].  Specific methods of distributing the points include, for example, minimizing the sum of all reciprocals of squares of distances between points.

==Infinitely many terms==

===Convergent series===

* A [[Sum-free sequence|sum-free sequence]] of increasing positive integers is one for which no number is the sum of any [[Subset|subset]] of the previous ones. The sum of the reciprocals of the numbers in any sum-free sequence is less than {{math|2.8570}}&nbsp;.

* The sum of the reciprocals of the heptagonal numbers converges to a known value that is not only [[Irrational number|irrational]] but also [[Transcendental number|transcendental]], and for which there exists a complicated formula.

* The sum of the reciprocals of the [[Twin prime|twin prime]]s, of which there may be finitely many or infinitely many, is known to be finite and is called [[Brun's constant]], approximately {{math|1.9022}}&nbsp;. The reciprocal of five conventionally appears twice in the sum.

* The sum of the reciprocals of the [[Proth prime]]s, of which there may be finitely many or infinitely many, is known to be finite, {{nobr|approximately {{math|0.747392479}} .}}<ref>{{cite journal |last1=Borsos |first1=Bertalan |last2=Kovács |first2=Attila |last3=Tihanyi |first3=Norbert |title=Tight upper and lower bounds for the reciprocal sum of Proth primes |journal=The Ramanujan Journal |date=1 September 2022 |volume=59 |issue=1 |pages=181–198 |doi=10.1007/s11139-021-00536-2|s2cid=246024152 |hdl=10831/83020 |hdl-access=free }}</ref>

* The [[Prime quadruplet|prime quadruplet]]s are pairs of twin primes with only one odd number between them. The sum of the reciprocals of the numbers in prime quadruplets is {{nobr| approximately {{math|0.8706}} .}}

* The sum of the reciprocals of the [[Perfect power#Examples and sums|perfect powers]] (including duplicates) {{nobr|is {{math|1}} .}}

* The sum of the reciprocals of the [[Perfect power#Examples and sums|perfect powers]] (excluding duplicates) is approximately {{math|0.8745}}&nbsp;.<ref>{{MathWorld|urlname=PerfectPower|title=Perfect Power}}</ref>

* The sum of the reciprocals of the powers <math>\ n^n\ </math> is approximately equal to {{math|1.2913}}&nbsp;. The sum is exactly equal to a definite [[Integral|integral]]: <math display="block">\ \sum_{n=1}^\infty \frac{1}{\ n^n } = \int_0^1 \frac{\ \mathrm{d}\ x\ }{\ x^x}\ </math>
: This identity was discovered by [[Biography:Johann Bernoulli|Johann Bernoulli]] in 1697, and is now known as one of the two [[Sophomore's dream]] identities.

* 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&nbsp;.
*The sum of the reciprocals of all the non-zero [[Triangular number#Other properties|triangular numbers]] {{nobr| is {{math|2}} .}}

* The [[Reciprocal Fibonacci constant|reciprocal Fibonacci constant]] is the sum of the reciprocals of the [[Fibonacci number]]s, which is known to be finite and irrational and approximately equal to 3.3599&nbsp;. For other finite sums of subsets of the reciprocals of Fibonacci numbers, see [[Fibonacci number#Reciprocal sums|here]].

* An [[Exponential factorial|exponential factorial]] is an operation [[Recurrence relation|recursively defined]] as <math>\ a_0 = 1, ~ a_n = n^{a_{n - 1}} ~.</math> For example, <math>\ a_4 = 4^{3^{2^{1}}}\ </math> 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|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&nbsp;.<ref>{{cite journal |last = Golomb |first = S.W. |year = 1970 |title = Powerful numbers |journal = [[American Mathematical Monthly]] |volume = 77 |issue = 8 |pages = 848–852 |doi = 10.2307/2317020 |jstor = 2317020}}</ref>

* The reciprocals of the [[Factorial#Series of reciprocals|factorials]] sum to the [[Transcendental number|transcendental number]] [[E (mathematical constant)|{{mvar|e}}]] (one of two constants called "Euler's number").

* The sum of the reciprocals of the [[Square number|square number]]s (the [[Basel problem]]) is the transcendental number {{nobr|{{math|{{sfrac|  π{{sup|2}} |6}} }},}} or {{math|''ζ''(2)}} where {{mvar|ζ}} is the [[Riemann zeta function]].

* The sum of the reciprocals of the cubes of positive integers is called [[Apéry's constant]] {{math|''ζ''(3)}}&nbsp;, and equals approximately 1.2021&nbsp;. This number is [[Irrational number|irrational]], but it is not known whether or not it is [[Transcendental number|transcendental]].

* The reciprocals of the non-negative integer powers of 2 sum to {{math|2}}&nbsp;.  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 {{mvar|a}} and the common ratio is {{mvar|r}} then the sum is {{nobr| {{math|{{sfrac|''r''|'' a'' (''r'' − 1) }} }} .}}

* The [[Kempner series]] is the sum of the reciprocals of all positive integers not containing the digit "9" in {{nobr|base {{math|10}} .}} Unlike the [[Harmonic series (mathematics)|harmonic series]], which does not exclude those numbers, this series converges, specifically to approximately {{math|22.9207}}&nbsp;.

* A [[Palindromic number|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 {{math|3.3703}}&nbsp;.

* A pentatope number is a number in the fifth cell of any row of [[Pascal's triangle]] starting with the five-term row {{math|1 4 6 4 1}}&nbsp;. The sum of the reciprocals of the pentatope numbers is {{sfrac|4| 3 }}&nbsp;.

* [[Sylvester's sequence]] is an [[Integer sequence|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 {{math| 2, 3, 7, 43, 1807}}&nbsp;. The sum of the reciprocals of the numbers in Sylvester's sequence {{nobr|is {{math|1}} .}}

* The [[Riemann zeta function]] {{math|''ζ''(''s'')}} is a [[Function (mathematics)|function]] of a complex variable {{mvar|s}} that [[Analytic continuation|analytically continues]] the sum of the infinite series  <math display="block">\ \sum_{n=1}^\infty \frac{1}{\ n^s\ } ~</math> to an analytic function on the entire complex plane except for ''s'' = 1, where {{math|''ζ''(''s'')}} has a pole. This series converges if and only if the real part of {{mvar|s}} is greater than {{math|1}}&nbsp;.

* The sum of the reciprocals of all the [[Fermat number]]s (numbers of the form <math>\ 2^{(2^n)} + 1\ </math>) {{OEIS|id=A051158}} is [[Irrational number|irrational]].

* The sum of the reciprocals of the [[Pronic number|pronic number]]s (products of two consecutive integers) (excluding&nbsp;{{math|0}}) is&nbsp;{{math|1}} (see [[Telescoping series]]).

===Divergent series===
* The {{mvar|n}}-th partial sum of the [[Harmonic series (mathematics)|harmonic series]], which is the sum of the reciprocals of the first {{mvar|n}} positive integers, diverges as {{mvar|n}} goes to infinity, albeit extremely slowly: The sum of the first {{math| {{10^|43}} }} terms is less than {{math|100}}&nbsp;. The difference between the cumulative sum and the [[Natural logarithm|natural logarithm]] of {{mvar|n}}  converges to the [[Euler–Mascheroni constant]], commonly denoted as <math>\ \gamma\ ,</math> which is approximately {{math|0.5772}}&nbsp;.

* The sum of the reciprocals of the primes diverges.

* The strong form of [[Dirichlet's theorem on arithmetic progressions]] implies that the sum of the reciprocals of the primes of the form {{nobr| {{math| 4 ''n'' + 3 }} }} is divergent.
*Similarly, the sum of the reciprocals of the primes of the form {{nobr| {{math| 4''n'' + 1 }} }} is divergent. By [[Fermat's theorem on sums of two squares]], it follows that the sum of reciprocals of numbers of the form <math>\ a^2 + b^2\ ,</math> where {{mvar|a}} and {{mvar|b}} are non-negative integers, not both equal to {{math|0}}, diverges, with or without repetition.

* If {{math|''a''(''k'')}} is any ascending series of positive integers with the property that there exists {{mvar|N}} such that {{math|''a''(''k'' + 1) − ''a''(''k'') < ''N''}} for all {{mvar|k}} then the sum of the reciprocals {{math|{{sfrac|1| ''a''(''k'') }} }} diverges.

* The [[Erdős conjecture on arithmetic progressions]]  states that if the sum of the reciprocals of the members of a set {{mvar|A}} of positive integers diverges, then {{mvar|A}} contains [[Arithmetic progression|arithmetic progression]]s of any length, however great. {{as of|2020}} the conjecture remains unproven.

==See also==
*[[Large set (combinatorics)|Large set]]
*Sum of squares
*[[Sums of powers]]

==References==
{{reflist|25em}}

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{{Sourceattribution|List of sums of reciprocals}}