Riemann Zeta Function at Even Integers

Theorem
The Riemann $\zeta$ function can be calculated for even integers as follows:

where:
 * $B_n$ are the Bernoulli numbers
 * $n$ is a positive integer.

Lemma
We also have:

Equating the coefficients of $(1)$ with the expression given in the lemma:


 * $\zeta \left({2 n}\right) = \left({-1}\right)^{n + 1} \dfrac {B_{2 n} 2^{2 n - 1} \pi^{2 n}} {\left({2 n}\right)!}$

Also see

 * Basel Problem
 * Riemann Zeta Function at Odd Integers: still unsolved