# Riemann Zeta Function at Even Integers/Examples/2

## Example of Riemann Zeta Function at Even Integers

The Riemann zeta function of $2$ is given by:

 $\ds \map \zeta 2$ $=$ $\ds \dfrac 1 {1^2} + \dfrac 1 {2^2} + \dfrac 1 {3^2} + \dfrac 1 {4^2} + \cdots$ $\ds$ $=$ $\ds \dfrac {\pi^2} 6$ $\ds$ $\approx$ $\ds 1 \cdotp 64493 \, 4066 \ldots$

## Proof

 $\ds \zeta \left({2}\right)$ $=$ $\ds \left({-1}\right)^2 \dfrac {B_2 2^1 \pi^2} {2!}$ Riemann Zeta Function at Even Integers $\ds$ $=$ $\ds \left({-1}\right)^2 \left({\dfrac 1 6}\right) \dfrac {2^1 \pi^2} {2!}$ Definition of Sequence of Bernoulli Numbers $\ds$ $=$ $\ds \left({\dfrac 1 6}\right) \left({\dfrac 2 2}\right) \pi^2$ Definition of Factorial $\ds$ $=$ $\ds \dfrac {\pi^2} 6$ simplifying

$\blacksquare$

The decimal expansion can be found by an application of arithmetic.