8 articles
1 + 2 + 3 + 4 + ⋯
The series 1 + 2 + 3 + 4 + ⋯, the sum of all natural numbers, diverges, but is assigned the value −1/12 in analytic continuation.
Apéry's constant
Apéry's constant, the value of the Riemann zeta function at 3, equals about 1.2020569 and is the sum of the reciprocals of the cubes of the positive integers.
Basel problem
The Basel problem, named for the Bernoulli family of Basel, asks for the exact sum of the reciprocals of the squares of the positive integers.
Class number formula
The class number formula expresses the residue at s = 1 of a number field's Dedekind zeta function through its class number, regulator, discriminant, and roots of unity.
Leibniz formula for π
The Leibniz formula for π, also called the Gregory–Leibniz series, is the alternating series 1 − 1/3 + 1/5 − 1/7 + ⋯ that sums to π/4.
List of mathematical series
The list of mathematical series, also covering partial sum formulae, collects the important named series such as the geometric, harmonic, and zeta series, with their convergence and sums.
Particular values of the Riemann zeta function
The particular values of the Riemann zeta function, also called zeta constants, are its values at integer arguments, such as ζ(2) = π²/6 and ζ(−1) = −1/12.
Ramanujan summation
Ramanujan summation is a technique by Srinivasa Ramanujan for assigning finite values to divergent series, famously giving 1 + 2 + 3 + ⋯ the value −1/12.