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]

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]

See also[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)]”. Template:ISO date/en閲覧。
  4. ^ Hofbauer, Josef. “[ A simple proof of 1+1/2^2+1/3^2+...=PI^2/6 and related identities]”. Template:ISO date/en閲覧。
  5. ^ Weisstein, Eric W., "Riemann Zeta Function" from MathWorld, equation 52
  6. ^ Abramowitz and Stegun