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We establish a connection between a function and a series representationusing 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.

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