This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums.

Sums of powers

See Faulhaber's formula. The first few values are: See zeta constants. The first few values are:

Power series

Low-order polylogarithms

Finite sums: Infinite sums, valid for |z|<1 (see polylogarithm): The following is a useful property to calculate low-integer-order polylogarithms recursively in closed form:

Exponential function

where T_{n}(z) is the Touchard polynomials.

Trigonometric, inverse trigonometric, hyperbolic, and inverse hyperbolic functions relationship

Modified-factorial denominators

Binomial coefficients

Harmonic numbers

(See harmonic numbers, themselves defined, and H(x) generalized to the real numbers)

Binomial coefficients

Trigonometric functions

Sums of sines and cosines arise in Fourier series.

Rational functions

Exponential function

Numeric series

These numeric series can be found by plugging in numbers from the series listed above.

Alternating harmonic series

Sum of reciprocal of factorials

Trigonometry and π

Reciprocal of tetrahedral numbers

Where

Exponential and logarithms