List of 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.

Sums of powers

$\sum _{i=1}^{n}i={\frac {n(n+1)}{2}}\,\!$
See also triangle number. This is one of the most useful series: many applications can be found throughout mathematics.
$\sum _{i=1}^{n}i^{2}={\frac {n(n+1)(2n+1)}{6}}={\frac {n^{3}}{3}}+{\frac {n^{2}}{2}}+{\frac {n}{6}}\,\!$
$\sum _{i=1}^{n}i^{3}=\left[{\frac {n(n+1)}{2}}\right]^{2}={\frac {n^{4}}{4}}+{\frac {n^{3}}{2}}+{\frac {n^{2}}{4}}=\left(\sum _{i=1}^{n}i\right)^{2}\,\!$
$\sum _{i=1}^{n}i^{4}={\frac {n(n+1)(2n+1)(3n^{2}+3n-1)}{30}}={\frac {6n^{5}+15n^{4}+10n^{3}-n}{30}}\,\!$
$\sum _{i=0}^{n}i^{s}={\frac {(n+1)^{s+1}}{s+1}}+\sum _{k=1}^{s}{\frac {B_{k}}{s-k+1}}{s \choose k}(n+1)^{s-k+1}\,\!$
Where $B_{k}\,$ is the $k\,$ th Bernoulli number, $B_{1}\,$ is negative and $s \choose k$ is the binomial coefficient (choose function).
$\sum _{i=1}^{\infty }i^{-s}=\prod _{p{\text{ prime}}}{\frac {1}{1-p^{-s}}}=\zeta (s)\,\!$

Where

\zeta (s)\,

is the Riemann zeta function.

Power series

Infinite sum (for $\|x\|<1$ )	Finite sum
$\sum _{i=0}^{\infty }x^{i}={\frac {1}{1-x}}\,\!$	$\sum _{i=0}^{n}x^{i}={\frac {1-x^{n+1}}{1-x}}=1+{\frac {1}{r}}\left(1-{\frac {1}{(1+r)^{n}}}\right)$ where $r>0$ and $x={\frac {1}{1+r}}.\,\!$
$\sum _{i=0}^{\infty }x^{2i}={\frac {1}{1-x^{2}}}\,\!$
$\sum _{i=1}^{\infty }ix^{i}={\frac {x}{(1-x)^{2}}}\,\!$	$\sum _{i=1}^{n}ix^{i}=x{\frac {1-x^{n}}{(1-x)^{2}}}-{\frac {nx^{n+1}}{1-x}}\,\!$
$\sum _{i=1}^{\infty }i^{2}x^{i}={\frac {x(1+x)}{(1-x)^{3}}}\,\!$	$\sum _{i=1}^{n}i^{2}x^{i}={\frac {x(1+x-(n+1)^{2}x^{n}+(2n^{2}+2n-1)x^{n+1}-n^{2}x^{n+2})}{(1-x)^{3}}}\,\!$
$\sum _{i=1}^{\infty }i^{3}x^{i}={\frac {x(1+4x+x^{2})}{(1-x)^{4}}}\,\!$
$\sum _{i=1}^{\infty }i^{4}x^{i}={\frac {x(1+x)(1+10x+x^{2})}{(1-x)^{5}}}\,\!$
$\sum _{i=1}^{\infty }i^{k}x^{i}=\operatorname {Li} _{-k}(x),\,\!$ where Li_s(x) is the polylogarithm of x.

Simple denominators

$\sum _{n=1}^{\infty }{\frac {x^{n}}{n}}=\log _{e}\left({\frac {1}{1-x}}\right)\quad {\mbox{ for }}|x|<1\!$

$\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}x^{2n+1}=x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}-\cdots =\arctan(x)\,\!$

$\sum _{n=0}^{\infty }{\frac {x^{2n+1}}{2n+1}}=\mathrm {arctanh} (x)\quad {\mbox{ for }}|x|<1\,\!$

$\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {\pi ^{2}}{6}}\,\!$
$\sum _{n=1}^{\infty }{\frac {1}{n^{4}}}={\frac {\pi ^{4}}{90}}\,\!$

$\sum _{n=1}^{\infty }{\frac {y}{n^{2}+y^{2}}}=-{\frac {1}{2y}}+{\frac {\pi }{2}}\coth(\pi y)$

Factorial denominators

Many power series which arise from Taylor's theorem have a coefficient containing a factorial.

$\sum _{i=0}^{\infty }{\frac {x^{i}}{i!}}=e^{x}$

$\sum _{i=0}^{\infty }i{\frac {x^{i}}{i!}}=xe^{x}$ (c.f. mean of Poisson distribution)
$\sum _{i=0}^{\infty }i^{2}{\frac {x^{i}}{i!}}=(x+x^{2})e^{x}$ (c.f. second moment of Poisson distribution)
$\sum _{i=0}^{\infty }i^{3}{\frac {x^{i}}{i!}}=(x+3x^{2}+x^{3})e^{x}$
$\sum _{i=0}^{\infty }i^{4}{\frac {x^{i}}{i!}}=(x+7x^{2}+6x^{3}+x^{4})e^{x}$

$\sum _{i=0}^{\infty }{\frac {(-1)^{i}}{(2i+1)!}}x^{2i+1}=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-\cdots =\sin x$

$\sum _{i=0}^{\infty }{\frac {(-1)^{i}}{(2i)!}}x^{2i}=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-\cdots =\cos x$

$\sum _{i=0}^{\infty }{\frac {x^{2i+1}}{(2i+1)!}}=\sinh x$

$\sum _{i=0}^{\infty }{\frac {x^{2i}}{(2i)!}}=\cosh x$

Modified-factorial denominators

$\sum _{n=0}^{\infty }{\frac {(2n)!}{4^{n}(n!)^{2}(2n+1)}}x^{2n+1}=\arcsin x\quad {\mbox{ for }}|x|<1\!$

$\sum _{i=0}^{\infty }{\frac {(-1)^{i}(2i)!}{4^{i}(i!)^{2}(2i+1)}}x^{2i+1}=\mathrm {arcsinh} (x)\quad {\mbox{ for }}|x|<1\!$

Binomial series

Geometric series:

$(1+x)^{-1}={\begin{cases}\displaystyle \sum _{i=0}^{\infty }(-x)^{i}&|x|<1\\\displaystyle \sum _{i=1}^{\infty }-(x)^{-i}&|x|>1\\\end{cases}}$

Binomial Theorem:

$(a+x)^{n}={\begin{cases}\displaystyle \sum _{i=0}^{\infty }{\binom {n}{i}}a^{n-i}x^{i}&|x|\!<\!|a|\\\displaystyle \sum _{i=0}^{\infty }{\binom {n}{i}}a^{i}x^{n-i}&|x|\!>\!|a|\\\end{cases}}$

$(1+x)^{\alpha }=\sum _{i=0}^{\infty }{\alpha \choose i}x^{i}\quad {\mbox{ for all }}|x|<1{\mbox{ and all complex }}\alpha \!$

with generalized binomial coefficients

{\alpha  \choose n}=\prod _{k=1}^{n}{\frac {\alpha -k+1}{k}}={\frac {\alpha (\alpha -1)\cdots (\alpha -n+1)}{n!}}\!

Square root:

${\sqrt {1+x}}=\sum _{i=0}^{\infty }{\frac {(-1)^{i}(2i)!}{(1-2i)i!^{2}4^{i}}}x^{i}\quad {\mbox{ for }}|x|<1\!$

Miscellaneous:

[1] $\sum _{i=0}^{\infty }{i+n \choose i}x^{i}={\frac {1}{(1-x)^{n+1}}}$
[1] $\sum _{i=0}^{\infty }{\frac {1}{i+1}}{2i \choose i}x^{i}={\frac {1}{2x}}(1-{\sqrt {1-4x}})$
[1] $\sum _{i=0}^{\infty }{2i \choose i}x^{i}={\frac {1}{\sqrt {1-4x}}}$
[1] $\sum _{i=0}^{\infty }{2i+n \choose i}x^{i}={\frac {1}{\sqrt {1-4x}}}\left({\frac {1-{\sqrt {1-4x}}}{2x}}\right)^{n}$

Binomial coefficients

$\sum _{i=0}^{n}{n \choose i}=2^{n}$
$\sum _{i=0}^{n}{n \choose i}a^{(n-i)}b^{i}=(a+b)^{n}$
$\sum _{i=0}^{n}(-1)^{i}{n \choose i}=0$
$\sum _{i=0}^{n}{i \choose k}={n+1 \choose k+1}$
$\sum _{i=0}^{n}{k+i \choose i}={k+n+1 \choose n}$
$\sum _{i=0}^{r}{r \choose i}{s \choose n-i}={r+s \choose n}$

Trigonometric functions

Sums of sines and cosines arise in Fourier series.

$\sum _{i=1}^{n}\sin \left({\frac {i\pi }{n}}\right)=0$
$\sum _{i=1}^{n}\cos \left({\frac {i\pi }{n}}\right)=0$

Unclassified

$\sum _{n=b+1}^{\infty }{\frac {b}{n^{2}-b^{2}}}=\sum _{n=1}^{2b}{\frac {1}{2n}}$

Related pages

Series (mathematics)
List of integrals
Summation
Taylor series
Binomial theorem
Gregory's series

Notes

Theoretical computer science cheat sheet

References

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

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[ctcs-1] Theoretical computer science cheat sheet