Riemann Zeta Function at Even Integers/Lemma/Proof 2

Proof
From Corollary to Laurent Series Expansion for Cotangent Function:


 * $\displaystyle \pi \cot \pi z = \frac 1 z - 2 \sum_{n \mathop = 1}^\infty \zeta \left({2 n}\right) z^{2 n - 1}$

where:
 * $z \in \C$ such that $\left\lvert{z}\right\rvert < 1$
 * $\zeta$ is the Riemann Zeta function.

Letting $x \in \R$ replace $z$, and multiplying through by $x$: