A short elementary proof of \(\displaystyle\sum_{k=1}^\infty\frac{1}{k^2}=\frac{\pi^2}{6}\)

Abstract

We give a short elementary proof of the well known identity \[\sum_{k=1}^\infty\frac{1}{k^2}=\frac{\pi^2}{6}.\] The idea is to write the partial sums of the series as a telescoping sum and to estimate the error term. The proof is only based on elementary recursion relations between the integrals \[ A_n=\int_0^{\frac{\pi}{2}}\cos^{2n}x\,dx \qquad\text{and}\qquad B_n=\int_0^{\frac{\pi}{2}}x^2\cos^{2n}x\,dx \] for \(n\geq 0\) and simple estimates.