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, $0^0$ is taken to have the value $1$
*$\$ denotes the fractional part of $x$
*$B_n(x)$ is a Bernoulli polynomial.
*$B_n$ is a Bernoulli number, and here, $B_1=-\frac.$
*$E_n$ is an Euler number.
*$\zeta(s)$ is the Riemann zeta function.
*$\Gamma(z)$ is the gamma function.
*$\psi_n(z)$ is a polygamma function.
*$\operatorname_s(z)$ is a polylogarithm.
*$n \choose k$ is binomial coefficient
*$\exp(x)$ denotes exponential of $x$

Sums of powers

See Faulhaber's formula.
*$\sum_^m k^=\frac$
The first few values are:
*$\sum_^m k=\frac$
*$\sum_^m k^2=\frac=\frac+\frac+\frac$
*\sum_^m k^3

=\left frac\right2=\frac+\frac+\frac

See zeta constants.
*$\zeta(2n)=\sum^_ \frac=(-1)^ \frac$
The first few values are:
*$\zeta(2)=\sum^_ \frac=\frac$ (the Basel problem)
*$\zeta(4)=\sum^_ \frac=\frac$
*$\zeta(6)=\sum^_ \frac=\frac$

Power series

Low-order polylogarithms

Finite sums:
*$\sum_^ z^k = \frac$, (geometric series)
*$\sum_^ z^k = \frac$
*$\sum_^ z^k = \frac-1 = \frac$
*$\sum_^n k z^k = z\frac$
*$\sum_^n k^2 z^k = z\frac$
*$\sum_^n k^m z^k = \left(z \frac\right)^m \frac$

Infinite sums, valid for $, z, <1$ (see polylogarithm):
*$\operatorname_n(z)=\sum_^ \frac$
The following is a useful property to calculate low-integer-order polylogarithms recursively in closed form:
*$\frac\operatorname_n(z)=\frac$
*$\operatorname_(z)=\sum_^\infty \frac=-\ln(1-z)$
*$\operatorname_(z)=\sum_^\infty z^k=\frac$
*$\operatorname_(z)=\sum_^\infty k z^k=\frac$
*$\operatorname_(z)=\sum_^\infty k^2 z^k=\frac$
*$\operatorname_(z)=\sum_^\infty k^3 z^k =\frac$
*$\operatorname_(z)=\sum_^\infty k^4 z^k =\frac$

Exponential function

*$\sum_^\infty \frac = e^z$
*$\sum_^\infty k\frac = z e^z$ (cf. mean of Poisson distribution)
*$\sum_^\infty k^2 \frac = (z + z^2) e^z$ (cf. second moment of Poisson distribution)
*$\sum_^\infty k^3 \frac = (z + 3z^2 + z^3) e^z$
*$\sum_^\infty k^4 \frac = (z + 7z^2 + 6z^3 + z^4) e^z$
*$\sum_^\infty k^n \frac = z \frac \sum_^\infty k^ \frac\,\! = e^z T_(z)$

where $T_(z)$ is the Touchard polynomials.

Trigonometric, inverse trigonometric, hyperbolic, and inverse hyperbolic functions relationship

*$\sum_^\infty \frac=\sin z$
*$\sum_^\infty \frac=\sinh z$
*$\sum_^\infty \frac=\cos z$
*$\sum_^\infty \frac=\cosh z$
*$\sum_^\infty \frac=\tan z, , z, <\frac$
*$\sum_^\infty \frac=\tanh z, , z, <\frac$
*$\sum_^\infty \frac=\cot z, , z, <\pi$
*$\sum_^\infty \frac=\coth z, , z, <\pi$
*$\sum_^\infty \frac=\csc z, , z, <\pi$
*$\sum_^\infty \frac=\operatorname z, , z, <\pi$
*$\sum_^\infty \frac=\operatorname z, , z, <\frac$
*$\sum_^\infty \frac=\sec z, , z, < \frac$
*$\sum_^\infty \frac=\operatornamez$ (versine)
*$\sum_^\infty \frac=\operatornamez$ (haversine

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