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.

Sums of powers

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

  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.

Power series

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.

Simple denominators

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

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

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

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

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

Factorial denominators

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)
• ${\displaystyle \sum _{i=0}^{\infty }i^{3}{\frac {x^{i}}{i!}}=(x+3x^{2}+x^{3})e^{x}}$
• ${\displaystyle \sum _{i=0}^{\infty }i^{4}{\frac {x^{i}}{i!}}=(x+7x^{2}+6x^{3}+x^{4})e^{x}}$

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

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

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

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

Modified-factorial denominators

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

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

Binomial series

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

Binomial coefficients

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

Trigonometric functions

Sums of sines and cosines arise in Fourier series.

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

Unclassified

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

Related pages

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