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

TL;DR

This paper introduces a novel proof technique connecting a function and series representations to derive exact values of the Riemann zeta function at even integers, generalizing Euler's approach to the Basel problem.

Contribution

It presents a new proof method for calculating z(2k) values, extending Euler's classical solution and providing a broader series framework.

Findings

Derived a general series from which z(2k) can be obtained as a limit

Proved the classical formula for z(2k) involving Bernoulli numbers

Established a connection similar to Euler's method for the Basel problem

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.