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.

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

  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

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 $x=\frac{1}{1+r}.\,\!$
$\sum_{i=0}^\infty x^{2i}= \frac{1}{1-x^2}\,\!$
$\sum_{i=1}^\infty i x^i = \frac{x}{(1-x)^2}\,\!$	$\sum_{i=1}^n i x^i = x\frac{1-x^n}{(1-x)^2} - \frac{n x^{n+1}}{1-x}\,\!$
$\sum_{i=1}^{\infty} i^2 x^i =\frac{x(1+x)}{(1-x)^3}\,\!$	$\sum_{i=1}^n i^2 x^i = \frac{x(1+x-(n+1)^2x^n+(2n^2+2n-1)x^{n+1}-n^2x^{n+2})}{(1-x)^3} \,\!$
$\sum_{i=1}^{\infty} i^3 x^i =\frac{x(1+4x+x^2)}{(1-x)^4}\,\!$
$\sum_{i=1}^{\infty} i^4 x^i =\frac{x(1+x)(1+10x+x^2)}{(1-x)^5}\,\!$
$\sum_{i=1}^{\infty} i^k x^i = \operatorname{Li}_{-k}(x),\,\!$ where Lis(x) is the polylogarithm of x.

Simple denominators

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

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

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

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

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

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

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

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

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:

• $\sqrt{1+x} = \sum_{i=0}^\infty \frac{(-1)^i(2i)!}{(1-2i)i!^24^i}x^i \quad\mbox{ for } |x|<1\!$

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

• $\sum_{n=b+1}^{\infty} \frac{b}{n^2 - b^2} = \sum_{n=1}^{2b} \frac{1}{2n}$

Other 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.