showing $-\eta(s) = \lim_{z \ \to \ -1} \sum_{n=1}^\infty z^{n} n^{-s}$

For $Re(s) > 0$, the Dirichlet eta function has the series representation $\eta(s) = \sum_{n=1}^\infty (-1)^{n+1} n^{-s}$. It extends analytically to the whole complex plane, and we have the following representation :
$$\boxed{\forall s \in \mathbb{C}, \qquad -\eta(s) = \lim_{z \ \to \ -1, |z| < 1} \sum_{n=1}^\infty z^{n} n^{-s}}$$

Question : do you know other proofs, preferably more general ?

Also, it should prove $f(s,e^{2i \pi x})$ is entire in $s$ whenever $x \in \mathbb{R}\setminus\mathbb{Z}$. Any ideas or references are welcome : extending $f(s,z)$ analytically beyond $|z| \le 1$, a functional equation for $f(s,z)$..

Solutions Collecting From Web of "showing $-\eta(s) = \lim_{z \ \to \ -1} \sum_{n=1}^\infty z^{n} n^{-s}$"