Convergent series

• 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 heptagonal numbers converges to a known value that is not only irrational but also transcendental, and for which there exists a complicated formula.

• 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 ${\displaystyle \ n^{n}\ }$ is approximately equal to 1.2913 . The sum is exactly equal to a definite integral:
  $${\displaystyle \ \sum _{n=1}^{\infty }{\frac {1}{\ n^{n}}}=\int _{0}^{1}{\frac {\ \mathrm {d} \ x\ }{\ x^{x}}}\ }$$

  

This identity was discovered by 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 .
• 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 ${\displaystyle \ a_{0}=1,~a_{n}=n^{a_{n-1}}~.}$ For example, ${\displaystyle \ a_{4}=4^{3^{2^{1}}}\ }$ 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 reciprocals of the factorials sum to the transcendental number e (one of two constants called "Euler's number").

• The sum of the reciprocals of the square numbers (the Basel problem) is the transcendental number   π² /6 , or ζ(2) where ζ is the Riemann zeta function.

• The sum of the reciprocals of the cubes of positive integers is called Apéry's constant ζ(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 .

• The Riemann zeta function ζ(s) is a function of a complex variable s that analytically continues the sum of the infinite series
  $${\displaystyle \ \sum _{n=1}^{\infty }{\frac {1}{\ n^{s}\ }}~}$$

  
  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 all the Fermat numbers (numbers of the form ${\displaystyle \ 2^{(2^{n})}+1\ }$) (sequence A051158 in the OEIS) is irrational.

• 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 ${\displaystyle \ \gamma \ ,}$ which is approximately 0.5772 .

• 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 4 n + 3 is divergent.
• 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 ${\displaystyle \ a^{2}+b^{2}\ ,}$ 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.

• The Erdős conjecture on arithmetic progressions states that if the sum of the reciprocals of the members of a set A of positive integers diverges, then A contains arithmetic progressions of any length, however great. As of 2020^[update] the conjecture remains unproven.