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List of mathematical seriesFrom Wikipedia, the free encyclopedia

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 [hide]

1 Sums of powers

2 Power series

2.1 Low-order polylogarithms

2.2 Exponential function

2.3 Trigonometric, inverse trigonometric, hyperbolic, and inverse hyperbolic functions

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2.4 Modified-factorial denominators

2.5 Binomial coefficients

2.6 Harmonic numbers

3 Binomial coefficients

4 Trigonometric functions

5 Rational functions

6 See also

7 Notes

8 References

Sums of powers [edit]See Faulhaber's formula.

The first few values are:

See zeta constants.

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The first few values are:

(the Basel problem)

Power series [edit]

Low-order polylogarithms [edit]

Finite sums:

, (geometric series)

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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]

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(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]

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Modified-factorial denominators [edit]

[1]

[1]

Binomial coefficients [edit]

(see Binomial theorem)

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[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]

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(see Multiset)

(see Vandermonde identity)

Trigonometric functions [edit]Sums of sines and cosines arise in Fourier series.

[3]

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[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.

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This page w as last modif ied on 21 June 2015, at 15:52.

See also [edit]Series (mathematics)

List of integrals

Summation

Taylor series

Binomial theorem

Gregory's series

On-Line Encyclopedia of Integer Sequences

Notes [edit]1. ^a b c d generatingfunctionology

2. ^a b c d Theoretical computer science cheat sheet

3. ^ "Bernoulli polynomials: Series representations (subsection 06/02)" . Retrieved 2 June 2011.

4. ^ Hofbauer, Josef. "A simple proof of 1+1/2 2^+1/3 2^+...=PI^ 2/6 and related identities" (PDF). Retrieved

2 June 2011.

5. ^ Weisstein, Eric W., "Riemann Zeta Function " from MathWorld, equation 52

6. ^ Abramowitz and Stegun

References [edit]Many books with a list of integrals also have a list of series.

Categories: Mathematical series Mathematics-related lists Mathematical tables

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