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]
Modified-factorial denominators[edit]
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 , and generalized to the real numbers)
Binomial coefficients[edit]
- (see Multiset)
- (see Vandermonde identity)
Trigonometric functions[edit]
Sums of sines and cosines arise in Fourier series.
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 tetrahedral numbers[edit]
Where
Exponential and logarithms[edit]
See also[edit]
Notes[edit]
- ^ Weisstein, Eric W. "Haversine". MathWorld. Wolfram Research, Inc. Archived from the original on 2005-03-10. Retrieved 2015-11-06.
- ^ a b c d Wilf, Herbert R. (1994). generatingfunctionology (PDF). Academic Press, Inc.
- ^ a b c d "Theoretical computer science cheat sheet" (PDF).
- ^
Calculate the Fourier expansion of the function on the interval :
- ^ "Bernoulli polynomials: Series representations (subsection 06/02)". Wolfram Research. Retrieved 2 June 2011.
- ^ Hofbauer, Josef. "A simple proof of 1 + 1/22 + 1/32 + ··· = π2/6 and related identities" (PDF). Retrieved 2 June 2011.
- ^ Sondow, Jonathan; Weisstein, Eric W. "Riemann Zeta Function (eq. 52)". MathWorld—A Wolfram Web Resource.
- ^ 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.
References[edit]
- Many books with a list of integrals also have a list of series.