Riemann Zeta Function at Even Integers

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


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

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

Proof
Let $|x| < 1$.

We also have:

Equating the coefficients of $(1)$ and $(2)$, we get:


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

Also see

 * Riemann Zeta Function at Odd Integers: still unsolved