Levi-Civita parallelogramoid
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 1.
- 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.
Sums of powers
See Faulhaber's formula.
The first few values are:
See zeta constants.
The first few values are:
- (the Basel problem)
Power series
Low-order polylogarithms
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
- (cf. mean of Poisson distribution)
- (cf. second moment of Poisson distribution)
where is the Touchard polynomials.
Trigonometric, inverse trigonometric, hyperbolic, and inverse hyperbolic functions
Modified-factorial denominators
Binomial coefficients
- (see Binomial theorem)
- [2]
- [2] , generating function of the Catalan numbers
- [2] , generating function of the Central binomial coefficients
Harmonic numbers
Binomial coefficients
- (see Multiset)
- (see Vandermonde identity)
Trigonometric functions
Sums of sines and cosines arise in Fourier series.
Rational functions
- An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition.[6] 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.
See also
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- Series (mathematics)
- List of integrals
- Summation
- Taylor series
- Binomial theorem
- Gregory's series
- On-Line Encyclopedia of Integer Sequences
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Notes
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References
- Many books with a list of integrals also have a list of series.