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.

  • Here, is taken to have the value
  • denotes the fractional part of
  • 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.
  • is binomial coefficient
  • denotes exponential of

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 relationship[edit]

  • (versine)
  • [1] (haversine)

Modified-factorial denominators[edit]

  • [2]
  • [2]

Binomial coefficients[edit]

  • (see Binomial theorem § Newton's generalized binomial theorem)
  • [3]
  • [3] , generating function of the Catalan numbers
  • [3] , generating function of the Central binomial coefficients
  • [3]

Harmonic numbers[edit]

(See harmonic numbers, themselves defined )

  • [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] as explained here. 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.

Exponential function[edit]

  • (see the Landsberg–Schaar relation)

Numeric series[edit]

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

Alternating harmonic series[edit]

Sum of reciprocal of factorials[edit]

Trigonometry and π[edit]

Reciprocal of triangular numbers[edit]


Reciprocal of tetrahedral numbers[edit]


Exponential and logarithms[edit]

See also[edit]


  1. ^ Weisstein, Eric W. "Haversine". MathWorld. Wolfram Research, Inc. Archived from the original on 2005-03-10. Retrieved 2015-11-06.
  2. ^ a b c d Wilf, Herbert R. (1994). generatingfunctionology (PDF). Academic Press, Inc.
  3. ^ a b c d "Theoretical computer science cheat sheet" (PDF).
  4. ^ Calculate the Fourier expansion of the function on the interval :
  5. ^ "Bernoulli polynomials: Series representations (subsection 06/02)". Wolfram Research. Retrieved 2 June 2011.
  6. ^ Hofbauer, Josef. "A simple proof of 1 + 1/22 + 1/32 + ··· = π2/6 and related identities" (PDF). Retrieved 2 June 2011.
  7. ^ Sondow, Jonathan; Weisstein, Eric W. "Riemann Zeta Function (eq. 52)". MathWorld—A Wolfram Web Resource.
  8. ^ Abramowitz, Milton; Stegun, Irene (1964). "6.4 Polygamma functions". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. p. 260. ISBN 0-486-61272-4.