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.

ums of powers

*$sum\_\{i=1\}^n\; i\; =\; frac\{n(n+1)\}\{2\}$
*$sum\_\{i=1\}^n\; i^2\; =\; frac\{n(n+1)(2n+1)\}\{6\}\; =\; frac\{n^3\}\{3\}\; +\; frac\{n^2\}\{2\}\; +\; frac\{n\}\{6\}$
*$sum\_\{i=1\}^n\; i^3\; =\; left(frac\{n(n+1)\}\{2\}\; ight)^2\; =\; frac\{n^4\}\{4\}\; +\; frac\{n^3\}\{2\}\; +\; frac\{n^2\}\{4\}\; =\; left\; [sum\_\{i=1\}^n\; i\; ight]\; ^2$
*$sum\_\{i=1\}^\{n\}\; i^\{4\}\; =\; frac\{n(n+1)(2n+1)(3n^\{2\}+3n-1)\}\{30\}$
*$sum\_\{i=0\}^n\; i^s\; =\; frac\{(n+1)^\{s+1\{s+1\}\; +\; sum\_\{k=1\}^sfrac\{B\_k\}\{s-k+1\}\{schoose\; k\}(n+1)^\{s-k+1\}$:where $B\_k$ is the "k"th Bernoulli number.
*$sum\_\{i=1\}^infty\; i^\{-s\}\; =\; prod\_\{p\; ext\{\; prime\; frac\{1\}\{1-p^\{-s\; =\; zeta(s)$:where $zeta(s)$ is the Reimann zeta function.

Power series

imple denominators

*$sum^\{infty\}\_\{i=1\}\; frac\{x^i\}i\; =\; log\_eleft(frac\{1\}\{1-x\}\; ight)\; quadmbox\{\; for\; \}\; |x|le\; 1,\; ,\; x\; ot=\; -1$

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

*

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

*

*

Binomial series

Binomial series (includes the square root for $alpha\; =\; 1/2$ and the infinite geometric series for $alpha\; =\; -1$):

Square root:
*

Geometric series:
*

General form:

*:with generalized binomial coefficients:: $\{alphachoose\; n\}\; =\; prod\_\{k=1\}^n\; frac\{alpha-k+1\}k\; =\; frac\{alpha(alpha-1)cdots(alpha-n+1)\}\{n!\}!$

* [http://www.tug.org/texshowcase/cheat.pdf Theoretical computer science cheat sheet] ] $sum\_\{i=0\}^infty\; \{i+n\; choose\; i\}\; x^i\; =\; frac\{1\}\{(1-x)^\{n+1$
* $sum\_\{i=0\}^infty\; frac\{1\}\{i+1\}\{2i\; choose\; i\}\; x^i\; =\; frac\{1\}\{2x\}(sqrt\{1-4x\})$
* $sum\_\{i=0\}^infty\; \{2i\; choose\; i\}\; x^i\; =\; frac\{1\}\{sqrt\{1-4x$
* $sum\_\{i=0\}^infty\; \{2i\; +\; n\; choose\; i\}\; x^i\; =\; frac\{1\}\{sqrt\{1-4xleft(frac\{1-sqrt\{1-4x\{2x\}\; ight)^n$

Binomial coefficients

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

Trigonometric functions

Sums of sines and cosines arise in Fourier series.

*$sum\_\{i=1\}^n\; sinleft(frac\{ipi\}\{n\}\; ight)\; =\; 0$
*$sum\_\{i=1\}^n\; cosleft(frac\{ipi\}\{n\}\; ight)\; =\; 0$

Unclassified

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

References

(Many books with a list of integrals also have a list of series.)

ee also

* Series (mathematics)
* List of integrals
* Summation
* Taylor series
* Binomial theorem
* Fourier series