List of mathematical series

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 powersEdit

See Faulhaber's formula.

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The first few values are:

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See zeta constants.

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The first few values are:

  •   (the Basel problem)
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Power seriesEdit

Low-order polylogarithmsEdit

Finite sums:

  •  , (geometric series)
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Infinite sums, valid for   (see polylogarithm):

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The following is a useful property to calculate low-integer-order polylogarithms recursively in closed form:

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Exponential functionEdit

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  •   (cf. mean of Poisson distribution)
  •   (cf. second moment of Poisson distribution)
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where   is the Touchard polynomials.

Trigonometric, inverse trigonometric, hyperbolic, and inverse hyperbolic functions relationshipEdit

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  •   (versine)
  •  [1] (haversine)
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Modified-factorial denominatorsEdit

  •  [2]
  •  [2]
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Binomial coefficientsEdit

  •   (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 numbersEdit

(See harmonic numbers, themselves defined  )

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  •  [2]
  •  [2]

Binomial coefficientsEdit

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  •   (see Multiset)
  •   (see Vandermonde identity)

Trigonometric functionsEdit

Sums of sines and cosines arise in Fourier series.

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  •  ,[4]
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  •  [5]
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  •  [6]
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Rational functionsEdit

  •  [7]
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  • 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.

Exponential functionEdit

  •  (see the Landsberg–Schaar relation)
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Numeric seriesEdit

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

Alternating harmonic seriesEdit

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Sum of reciprocal of factorialsEdit

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Trigonometry and πEdit

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Reciprocal of triangular numbersEdit

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Where  

Reciprocal of tetrahedral numbersEdit

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Where  

Exponential and logarithmsEdit

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See alsoEdit

NotesEdit

  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.

ReferencesEdit