# List of mathematical series

From Wikipedia

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

- Here, is taken to have the value 1.
- is a Bernoulli polynomial.
- is a Bernoulli number, and here,
- is an Euler number.
- is the Riemann zeta function.
- is the gamma function.
- is a polygamma function.
- is a polylogarithm.

## Contents

## Sums of powers[edit]

See Faulhaber's formula.

The first few values are:

See zeta constants.

The first few values are:

- (the Basel problem)

## Power series[edit]

### Low-order polylogarithms[edit]

Finite sums:

- , (geometric series)

Infinite sums, valid for (see polylogarithm):

The following is a useful property to calculate low-integer-order polylogarithms recursively in closed form:

### Exponential function[edit]

- (cf. mean of Poisson distribution)

- (cf. second moment of Poisson distribution)

where is the Touchard polynomials.

### Trigonometric, inverse trigonometric, hyperbolic, and inverse hyperbolic functions[edit]

### Modified-factorial denominators[edit]

^{[1]}

^{[1]}

### Binomial coefficients[edit]

- (see Binomial theorem)
^{[2]}

^{[2]}, generating function of the Catalan numbers

^{[2]}, generating function of the Central binomial coefficients

^{[2]}

### Harmonic numbers[edit]

^{[1]}

^{[1]}

## Binomial coefficients[edit]

- (see Multiset)

- (see Vandermonde identity)

## Trigonometric functions[edit]

Sums of sines and cosines arise in Fourier series.

^{[3]}

^{[4]}

## Rational functions[edit]

^{[5]}

- An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition.
^{[6]}This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.

## See also[edit]

## Notes[edit]

- ^
^{a}^{b}^{c}^{d}generatingfunctionology - ^
^{a}^{b}^{c}^{d}Theoretical computer science cheat sheet **^**"Bernoulli polynomials: Series representations (subsection 06/02)". Retrieved 2 June 2011.**^**Hofbauer, Josef. "A simple proof of 1+1/2^2+1/3^2+...=PI^2/6 and related identities". Retrieved 2 June 2011.**^**Weisstein, Eric W., "Riemann Zeta Function" from MathWorld, equation 52**^**Abramowitz and Stegun

## References[edit]

- Many books with a list of integrals also have a list of series.