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List of mathematical series

From 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
  • 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

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See Faulhaber's formula.

The first few values are:

See zeta constants.

The first few values are:

  • (the Basel problem)

Power series

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Low-order polylogarithms

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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

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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 relationship

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  • (versine)
  • [1] (haversine)

Modified-factorial denominators

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  • [2]
  • [2]

Binomial coefficients

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  • (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

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(See harmonic numbers, themselves defined , and generalized to the real numbers)

  • [2]
  • [2]

Binomial coefficients

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  • (see Multiset)
  • (see Vandermonde identity)

Trigonometric functions

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Sums of sines and cosines arise in Fourier series.

  • ,[4]
  • [5]
  • [6]

Rational functions

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  • [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

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  • (see the Landsberg–Schaar relation)

Numeric series

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

Alternating harmonic series

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Sum of reciprocal of factorials

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Trigonometry and π

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Reciprocal of tetrahedral numbers

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Where

Exponential and logarithms

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  • , that is

See also

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Notes

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  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. Courier Corporation. p. 260. ISBN 0-486-61272-4.

References

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