A new proof for the exact values of $\zeta (2k)$ for $k\in \mathbb{N}$

We establish a connection between a function and a series representation using a similar technique with that that Euler used to solve the Basel problem. Our result concerns a more general series from which one can obtain $\zeta (2k)$ as a limit case. We also are able to prove the well known result expressing $\zeta (2k)$ with Bernoulli numbers as an application.