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.
$B_n(x)$ is a Bernoulli polynomial.
$B_n$ is a Bernoulli number, and here, $B_1=-\frac{1}{2}.$
$E_n$ is an Euler number.
$\zeta(s)$ is the Riemann zeta function.
$\Gamma(z)$ is the gamma function.
$\psi_n(z)$ is a polygamma function.
$\operatorname{Li}_s(z)$ is a polylogarithm.

Sums of powers

See Faulhaber's formula.

$\sum_{k=0}^m k^{n-1}=\frac{B_n(m+1)-B_n}{n}\,\!$

The first few values are:

$\sum_{k=1}^m k=\frac{m(m+1)}{2}\,\!$

$\sum_{k=1}^m k^2=\frac{m(m+1)(2m+1)}{6}=\frac{m^3}{3}+\frac{m^2}{2}+\frac{m}{6}\,\!$

$\sum_{k=1}^m k^3 =\left[\frac{m(m+1)}{2}\right]^2=\frac{m^4}{4}+\frac{m^3}{2}+\frac{m^2}{4}\,\!$

See zeta constants.

$\zeta(2n)=\sum^{\infty}_{k=1} \frac{1}{k^{2n}}=(-1)^{n+1} \frac{B_{2n} (2\pi)^{2n}}{2(2n)!}$

The first few values are:

$\zeta(2)=\sum^{\infty}_{k=1} \frac{1}{k^2}=\frac{\pi^2}{6}\,\!$ (the Basel problem)

$\zeta(4)=\sum^{\infty}_{k=1} \frac{1}{k^4}=\frac{\pi^4}{90}\,\!$

$\zeta(6)=\sum^{\infty}_{k=1} \frac{1}{k^6}=\frac{\pi^6}{945}\,\!$

Power series

Low-order polylogarithms

Finite sums:

$\sum_{k=0}^{n} z^k = \frac{1-z^{n+1}}{1-z}\,\!$ , (geometric series)

$\sum_{k=1}^n k z^k = z\frac{1-(n+1)z^n+nz^{n+1}}{(1-z)^2}\,\!$

$\sum_{k=1}^n k^2 z^k = z\frac{1+z-(n+1)^2z^n+(2n^2+2n-1)z^{n+1}-n^2z^{n+2}}{(1-z)^3} \,\!$

$\sum_{k=1}^n k^m z^k = \left(z \frac{d}{dz}\right)^m \frac{z-z^{n+1}}{1-z}$

Infinite sums, valid for $|z|<1$ (see polylogarithm):

$\operatorname{Li}_n(z)=\sum_{k=1}^{\infty} \frac{z^k}{k^n}\,\!$

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

$\frac{d}{dz}\operatorname{Li}_n(z)=\frac{\operatorname{Li}_{n-1}(z)}{z}\,\!$

$\operatorname{Li}_{1}(z)=\sum_{k=1}^\infty \frac{z^k}{k}=-\ln(1-z)\!$

$\operatorname{Li}_{0}(z)=\sum_{k=1}^\infty z^k=\frac{z}{1-z}\!$

$\operatorname{Li}_{-1}(z)=\sum_{k=1}^\infty k z^k=\frac{z}{(1-z)^2}\,\!$

$\operatorname{Li}_{-2}(z)=\sum_{k=1}^\infty k^2 z^k=\frac{z(1+z)}{(1-z)^3}\,\!$

$\operatorname{Li}_{-3}(z)=\sum_{k=1}^\infty k^3 z^k =\frac{z(1+4z+z^2)}{(1-z)^4}\,\!$

$\operatorname{Li}_{-4}(z)=\sum_{k=1}^\infty k^4 z^k =\frac{z(1+z)(1+10z+z^2)}{(1-z)^5}\,\!$

Exponential function

$\sum_{k=0}^\infty \frac{z^k}{k!} = e^z\,\!$

$\sum_{k=0}^\infty k\frac{z^k}{k!} = z e^z\,\!$ (cf. mean of Poisson distribution)

$\sum_{k=0}^\infty k^2 \frac{z^k}{k!} = (z + z^2) e^z\,\!$ (cf. second moment of Poisson distribution)

$\sum_{k=0}^\infty k^3 \frac{z^k}{k!} = (z + 3z^2 + z^3) e^z\,\!$

$\sum_{k=0}^\infty k^4 \frac{z^k}{k!} = (z + 7z^2 + 6z^3 + z^4) e^z\,\!$

$\sum_{k=0}^\infty k^n \frac{z^k}{k!} = z \frac{d}{dz} \sum_{k=0}^\infty k^{n-1} \frac{z^k}{k!}\,\! = e^z T_{n}(z)$

where $T_{n}(z)$ is the Touchard polynomials.

Trigonometric, inverse trigonometric, hyperbolic, and inverse hyperbolic functions

$\sum_{k=0}^\infty \frac{(-1)^k z^{2k+1}}{(2k+1)!}=\sin z\,\!$

$\sum_{k=0}^\infty \frac{z^{2k+1}}{(2k+1)!}=\sinh z\,\!$

$\sum_{k=0}^\infty \frac{(-1)^k z^{2k}}{(2k)!}=\cos z\,\!$

$\sum_{k=0}^\infty \frac{z^{2k}}{(2k)!}=\cosh z\,\!$

$\sum_{k=1}^\infty \frac{(-1)^{k-1}(2^{2k}-1)2^{2k}B_{2k}z^{2k-1}}{(2k)!}=\tan z, |z|<\frac{\pi}{2}\,\!$

$\sum_{k=1}^\infty \frac{(2^{2k}-1)2^{2k}B_{2k}z^{2k-1}}{(2k)!}=\tanh z, |z|<\frac{\pi}{2}\,\!$

$\sum_{k=0}^\infty \frac{(-1)^k2^{2k}B_{2k}z^{2k-1}}{(2k)!}=\cot z, |z|<\pi\,\!$

$\sum_{k=0}^\infty \frac{2^{2k}B_{2k}z^{2k-1}}{(2k)!}=\coth z, |z|<\pi\,\!$

$\sum_{k=0}^\infty \frac{(-1)^{k-1}(2^{2k}-2)B_{2k}z^{2k-1}}{(2k)!}=\csc z, |z|<\pi\,\!$

$\sum_{k=0}^\infty \frac{-(2^{2k}-2)B_{2k}z^{2k-1}}{(2k)!}=\operatorname{csch} z, |z|<\pi\,\!$

$\sum_{k=0}^\infty \frac{(-1)^kE_{2k}z^{2k}}{(2k)!}=\sec z, |z|<\frac{\pi}{2}\,\!$

$\sum_{k=0}^\infty \frac{E_{2k}z^{2k}}{(2k)!}=\operatorname{sech} z, |z|<\frac{\pi}{2}\,\!$

$\sum_{k=0}^\infty \frac{(2k)!z^{2k+1}}{2^{2k}(k!)^2(2k+1)}=\arcsin z, |z|\le1\,\!$

$\sum_{k=0}^\infty \frac{(-1)^k(2k)!z^{2k+1}}{2^{2k}(k!)^2(2k+1)}=\operatorname{arcsinh} {z}, |z|\le1\,\!$

$\sum_{k=0}^\infty \frac{(-1)^kz^{2k+1}}{2k+1}=\arctan z, |z|<1\,\!$

$\sum_{k=0}^\infty \frac{z^{2k+1}}{2k+1}=\operatorname{arctanh} z, |z|<1\,\!$

$\ln2+\sum_{k=1}^\infty \frac{(-1)^{k-1}(2k)!z^{2k}}{2^{2k+1}k(k!)^2}=\ln\left(1+\sqrt{1+z^2}\right), |z|\le1\,\!$

Modified-factorial denominators

$\sum^{\infty}_{k=0} \frac{(4k)!}{2^{4k} \sqrt{2} (2k)! (2k+1)!} z^k = \sqrt{\frac{1-\sqrt{1-z}}{z}}, |z|<1$ ^[1]

$\sum^{\infty}_{k=0} \frac{2^{2k} (k!)^2}{(k+1) (2k+1)!} z^{2k+2} = \left(\arcsin{z}\right)^2, |z|\le1$ ^[1]

$\sum^{\infty}_{n=0} \frac{\prod_{k=0}^{n-1}(4k^2+\alpha^2)}{(2n)!} z^{2n} + \sum^{\infty}_{n=0} \frac{\alpha \prod_{k=0}^{n-1}[(2k+1)^2+\alpha^2]}{(2n+1)!} z^{2n+1} = e^{\alpha \arcsin{z}}, |z|\le1$

Binomial coefficients

$(1+z)^\alpha = \sum_{k=0}^\infty {\alpha \choose k} z^k, |z|<1$ (see Binomial theorem)
^[2] $\sum_{k=0}^\infty {{\alpha+k-1} \choose k} z^k = \frac{1}{(1-z)^\alpha}, |z|<1$

^[2] $\sum_{k=0}^\infty \frac{1}{k+1}{2k \choose k} z^k = \frac{1-\sqrt{1-4z}}{2z}, |z|\leq\frac{1}{4}$ , generating function of the Catalan numbers

^[2] $\sum_{k=0}^\infty {2k \choose k} z^k = \frac{1}{\sqrt{1-4z}}, |z|<\frac{1}{4}$ , generating function of the Central binomial coefficients

^[2] $\sum_{k=0}^\infty {2k + \alpha \choose k} z^k = \frac{1}{\sqrt{1-4z}}\left(\frac{1-\sqrt{1-4z}}{2z}\right)^\alpha, |z|<\frac{1}{4}$

Harmonic numbers

$\sum_{k=1}^\infty H_k z^k = \frac{-\ln(1-z)}{1-z}, |z|<1$

$\sum_{k=1}^\infty \frac{H_k}{k+1} z^{k+1} = \frac{1}{2}\left[\ln(1-z)\right]^2, \qquad |z|<1$

$\sum_{k=1}^\infty \frac{(-1)^{k-1} H_{2k}}{2k+1} z^{2k+1} = \frac{1}{2} \arctan{z} \log{(1+z^2)}, \qquad |z|<1$ ^[1]

$\sum_{n=0}^\infty \sum_{k=0}^{2n} \frac{(-1)^k}{2k+1} \frac{z^{4n+2}}{4n+2} = \frac{1}{4} \arctan{z} \log{\frac{1+z}{1-z}},\qquad |z|<1$ ^[1]

Binomial coefficients

$\sum_{k=0}^n {n \choose k} = 2^n$

$\sum_{k=0}^n (-1)^k {n \choose k} = 0$

$\sum_{k=0}^n {k \choose m} = { n+1 \choose m+1 }$

$\sum_{k=0}^n {m+k-1 \choose k} = { n+m \choose n }$ (see Multiset)

$\sum_{k=0}^n {\alpha \choose k}{\beta \choose n-k} = {\alpha+\beta \choose n}$ (see Vandermonde identity)

Trigonometric functions

Sums of sines and cosines arise in Fourier series.

$\sum_{k=1}^\infty \frac{\sin(k\theta)}{k}=\frac{\pi-\theta}{2}, 0<\theta<2\pi\,\!$

$\sum_{k=1}^\infty \frac{\cos(k\theta)}{k}=-\frac{1}{2}\ln(2-2\cos\theta), \theta\in\mathbb{R}\,\!$

$\sum_{k=0}^\infty \frac{\sin[(2k+1)\theta]}{2k+1}=\frac{\pi}{4}, 0<\theta<\pi\,\!$

$B_n(x)=-\frac{n!}{2^{n-1}\pi^n}\sum_{k=1}^\infty \frac{1}{k^n}\cos\left(2\pi kx-\frac{\pi n}{2}\right), 0<x<1\,\!$ ^[3]

$\sum_{k=0}^n \sin(\theta+k\alpha)=\frac{\sin\frac{(n+1)\alpha}{2}\sin(\theta+\frac{n\alpha}{2})}{\sin\frac{\alpha}{2}}\,\!$

$\sum_{k=1}^{n-1} \sin\frac{\pi k}{n}=\cot\frac{\pi}{2n}\,\!$

$\sum_{k=1}^{n-1} \sin\frac{2\pi k}{n}=0\,\!$

$\sum_{k=0}^{n-1} \csc^2\left(\theta+\frac{\pi k}{n}\right)=n^2\csc^2(n\theta)\,\!$ ^[4]

$\sum_{k=1}^{n-1} \csc^2\frac{\pi k}{n}=\frac{n^2-1}{3}\,\!$

$\sum_{k=1}^{n-1} \csc^4\frac{\pi k}{n}=\frac{n^4+10n^2-11}{45}\,\!$

Rational functions

$\sum_{m=b+1}^{\infty} \frac{b}{m^2 - b^2} = \frac{1}{2} H_{2b}$

$\sum^{\infty}_{m=1} \frac{y}{m^2+y^2} = -\frac{1}{2y}+\frac{\pi}{2}\coth(\pi y)$ ^[5]

An infinite series of any rational function of $n$ 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

Series (mathematics)
List of integrals
Summation
Taylor series
Binomial theorem
Gregory's series
On-Line Encyclopedia of Integer Sequences

Notes

↑ 1.0 1.1 1.2 1.3 generatingfunctionology
↑ 2.0 2.1 2.2 2.3 Theoretical computer science cheat sheet
↑ "Bernoulli polynomials: Series representations (subsection 06/02)". Retrieved 2 June 2011.
↑ Hofbauer, Josef. "A simple proof of 1+1/2^2+1/3^2+...=PI^2/6 and related identities". Retrieved 2 June 2011.
↑ Weisstein, Eric W., "Riemann Zeta Function" from MathWorld, equation 52
↑ Abramowitz and Stegun

References

Many books with a list of integrals also have a list of series.