List of series

Template:Orphan 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.

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

  Where $B_k$ is the $k$th Bernoulli number, $B_1$ is negative and $(\genfrac{}{}{0ex}{}{s}{k})$ is the binomial coefficient (choose function).

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

Where $\zeta (s)$ is the Riemann zeta function.

Power series

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

Simple denominators

• $\sum _{n=1}^{\mathrm{\infty }}\frac{x^n}{n}={\log }_e\left(\frac{1}{1-x}\right)\text{ for }|x|<1$

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

• $\sum _{n=0}^{\mathrm{\infty }}\frac{x^{2n+1}}{2n+1}=\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}(x)\text{ for }|x|<1$

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

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

Factorial denominators

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

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

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

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

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

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

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

Modified-factorial denominators

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

• $\sum _{i=0}^{\mathrm{\infty }}\frac{(-1{)}^i(2i)!}{4^i(i!{)}^2(2i+1)}{x}^{2i+1}=\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(x)\text{ for }|x|<1$

Binomial series

Geometric series:

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

Binomial Theorem:

• $(a+x{)}^n=\{\begin{array}{cc}\sum _{i=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{n}{i})a^{n-i}{x}^i& |x|<|a|\\ \sum _{i=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{n}{i})a^i{x}^{n-i}& |x|>|a|\end{array}$

• $(1+x{)}^{\alpha }=\sum _{i=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{\alpha }{i})x^i\text{ for all }|x|<1\text{ and all complex }\alpha$

with generalized binomial coefficients

$(\genfrac{}{}{0ex}{}{\alpha }{n})=\prod _{k=1}^n\frac{\alpha -k+1}{k}=\frac{\alpha (\alpha -1)\cdots (\alpha -n+1)}{n!}$

Square root:

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

Miscellaneous:

• $\sum _{i=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{i+n}{i})x^i=\frac{1}{(1-x{)}^{n+1}}$
• $\sum _{i=0}^{\mathrm{\infty }}\frac{1}{i+1}(\genfrac{}{}{0ex}{}{2i}{i})x^i=\frac{1}{2x}(1-\sqrt{1-4x})$
• $\sum _{i=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{2i}{i})x^i=\frac{1}{\sqrt{1-4x}}$
• $\sum _{i=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{2i+n}{i})x^i=\frac{1}{\sqrt{1-4x}}{\left(\frac{1-\sqrt{1-4x}}{2x}\right)}^n$

Binomial coefficients

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

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

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

Related pages

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

References

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