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.

• ${\displaystyle \sum _{i=1}^{n}i={\frac {n(n+1)}{2}\,\!}$

  See also triangle number. This is one of the most useful series: many applications can be found throughout mathematics.

• ${\displaystyle \sum _{i=1}^{n}i^{2}={\frac {n(n+1)(2n+1)}{6}={\frac {n^{3}{3}+{\frac {n^{2}{2}+{\frac {n}{6}\,\!}$
• ${\displaystyle \sum _{i=1}^{n}i^{3}=\left[{\frac {n(n+1)}{2}\right]^{2}={\frac {n^{4}{4}+{\frac {n^{3}{2}+{\frac {n^{2}{4}=\left(\sum _{i=1}^{n}i\right)^{2}\,\!}$
• ${\displaystyle \sum _{i=1}^{n}i^{4}={\frac {n(n+1)(2n+1)(3n^{2}+3n-1)}{30}={\frac {6n^{5}+15n^{4}+10n^{3}-n}{30}\,\!}$
• ${\displaystyle \sum _{i=0}^{n}i^{s}={\frac {(n+1)^{s+1}{s+1}+\sum _{k=1}^{s}{\frac {B_{k}{s-k+1}{s \choose k}(n+1)^{s-k+1}\,\!}$

  Where ${\displaystyle B_{k}\,}$ is the ${\displaystyle k\,}$th Bernoulli number, ${\displaystyle B_{1}\,}$ is negative and ${\displaystyle s \choose k}$ is the binomial coefficient (choose function).

• ${\displaystyle \sum _{i=1}^{\infty }i^{-s}=\prod _{p{\text{ prime}{\frac {1}{1-p^{-s}=\zeta (s)\,\!}$

Where ${\displaystyle \zeta (s)\,}$ is the Riemann zeta function.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Geometric series:

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

Binomial Theorem:

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

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

with generalized binomial coefficients

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

Square root:

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

Miscellaneous:

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

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

Sums of sines and cosines arise in Fourier series.

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

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

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

1. ↑ ^{1.0 1.1 1.2 1.3} Theoretical computer science cheat sheet

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