Mittag-Leffler Expansion for Cotangent Function/Proof 2

Proof
Let $\zeta\left(s\right)$ be the Riemann zeta function.

Let $\displaystyle g\left(z\right) = \sum_{n\mathop=1}^\infty z^n \zeta\left(2n\right)$ be the generating function of $\zeta\left(2n\right)$

By Power Series Expansion for Cotangent Function, for $\left\vert z \right\vert < 1$:

By Riemann Zeta Function at Even Integers:

Thus:

By Analytic Continuation of Generating Function of Dirichlet Series and Uniqueness of Analytic Continuation:

for all of $\C$, as this is the overlap of their domains.

Thus: