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

Abstract: 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.