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
*\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{6 n^5+15 n^4+10 n^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
### Simple denominators
*\sum^{\infty}_{n=1} \frac{x^n}n = \log_e\left(\frac{1}{1-x}\right) \quad\mbox{ for } |x| 

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

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

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

*\sum^{\infty}_{n=1} \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^{\infty}_{i=0} \frac{x^i}{i!} = e^x

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

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

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

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

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

### Modified-factorial denominators
*\sum^{\infin}_{n=0} \frac{(2n)!}{4^n (n!)^2 (2n+1)} x^{2n+1} = \arcsin x\quad\mbox{ for } |x| 

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

### Binomial series 
Geometric series:
*
 (1+x)^{-1} = 
 \begin{cases} 
   \displaystyle  \sum_{i=0}^\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|  \\
 \end{cases}

*(1+x)^\alpha = \sum_{i=0}^\infty {\alpha \choose i} x^i\quad\mbox{ for all } |x| 
::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!^24^i}x^i \quad\mbox{ for } |x|

Miscellaneous:
*  \sum_{i=0}^\infty {i+n \choose i} x^i = \frac{1}{(1-x)^{n+1}}
* \sum_{i=0}^\infty \frac{1}{i+1}{2i \choose i} x^i = \frac{1}{2x}(1-\sqrt{1-4x})
* \sum_{i=0}^\infty {2i \choose i} x^i = \frac{1}{\sqrt{1-4x}}
* \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
## References
*Many books with a list of integrals also have a list of series.