A simple computation of $\zeta(2k)$ by using Bernoulli polynomials and a telescoping series

TL;DR

This paper introduces a straightforward calculus-based method using Bernoulli polynomials and telescoping series to compute the values of the Riemann zeta function at even integers, simplifying previous proofs.

Contribution

It provides a new, elementary proof of Euler's formulas for z(2k) using only basic properties of Bernoulli polynomials and telescoping series.

Findings

Derives z(2k) using a telescoping series

Extends method to z(2k+1) and harmonic numbers

Provides integral formulas for these values

Abstract

We present a new proof of Euler's formulas for $\zeta(2k)$, where $k = 1,2,3,...$, which uses only the defining properties of the Bernoulli polynomials, obtaining the value of $\zeta(2k)$ by summing a telescoping series. Only basic techniques from Calculus are needed to carry out the computation. The method also applies to $\zeta(2k+1)$ and the harmonic numbers, yielding integral formulas for these.