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 rendered as
This can also be seen rendered in the elegant form:


 * $\zeta \left({r}\right) = \dfrac 1 2 \left\vert{B_r}\right\vert \dfrac {\left({2 \pi}\right)^r} {r!}$

for $r = 2 n$, $n \ge 1$.

Also see

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