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

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.