List of mathematical series

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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[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]

  • (versine)
  • [1] (haversine)

Modified-factorial denominators[edit]

  • [2]
  • [2]

Binomial coefficients[edit]

  • (see Binomial theorem)
  • [3]
  • [3] , generating function of the Catalan numbers
  • [3] , generating function of the Central binomial coefficients
  • [3]

Harmonic numbers[edit]

  • [2]
  • [2]

Binomial coefficients[edit]

  • (see Multiset)
  • (see Vandermonde identity)

Trigonometric functions[edit]

Sums of sines and cosines arise in Fourier series.

  • [4]
  • [5]
  • [6]

Rational functions[edit]

  • [7]
  • An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition.[8] 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]