This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums.

• Here, ${\displaystyle 0^{0}}$ is taken to have the value ${\displaystyle 1}$
• ${\displaystyle \{x\}}$ denotes the fractional part of ${\displaystyle x}$
• ${\displaystyle B_{n}(x)}$ is a Bernoulli polynomial.
• ${\displaystyle B_{n}}$ is a Bernoulli number, and here, ${\displaystyle B_{1}=-{\frac {1}{2}}.}$
• ${\displaystyle E_{n}}$ is an Euler number.
• ${\displaystyle \zeta (s)}$ is the Riemann zeta function.
• ${\displaystyle \Gamma (z)}$ is the gamma function.
• ${\displaystyle \psi _{n}(z)}$ is a polygamma function.
• ${\displaystyle \operatorname {Li} _{s}(z)}$ is a polylogarithm.
• ${\displaystyle n \choose k}$ is binomial coefficient
• ${\displaystyle \exp(x)}$ denotes exponential of ${\displaystyle x}$

Sums of powers

See Faulhaber's formula.

• ${\displaystyle \sum _{k=0}^{m}k^{n-1}={\frac {B_{n}(m+1)-B_{n}}{n}}}$

The first few values are:

• ${\displaystyle \sum _{k=1}^{m}k={\frac {m(m+1)}{2}}}$
• ${\displaystyle \sum _{k=1}^{m}k^{2}={\frac {m(m+1)(2m+1)}{6}}={\frac {m^{3}}{3}}+{\frac {m^{2}}{2}}+{\frac {m}{6}}}$
• ${\displaystyle \sum _{k=1}^{m}k^{3}=\left^{2}={\frac {m^{4}}{4}}+{\frac {m^{3}}{2}}+{\frac {m^{2}}{4}}}$

See zeta constants.

• ${\displaystyle \zeta (2n)=\sum _{k=1}^{\infty }{\frac {1}{k^{2n}}}=(-1)^{n+1}{\frac {B_{2n}(2\pi )^{2n}}{2(2n)!}}}$

The first few values are:

• ${\displaystyle \mathrm{\zeta <!--\; \zeta \; -->}(2)=\underset{k=1}{\overset{\mathrm{\infty <!--\; \infty \; -->}}{\sum <!--\; \sum \; -->}}\frac{1}{k^2}=\frac{{\mathrm{\pi <!--\; \pi \; -->}}^2}{6}}$Zdroj:https://en.wikipedia.org?pojem=List_of_mathematical_series
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