A short elementary proof of ${\displaystyle \sum _{k=1}^{\mathrm{\infty }}\frac{1}{k^2}=\frac{{\pi }^2}{6}}$

Abstract

We give a short elementary proof of the well known identity $$\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k^2}=\frac{{\pi }^2}{6}.$$ 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 $$A_n={\int }_0^{\frac{\pi }{2}}{\cos }^{2n}xdx\text{and}{B}_n={\int }_0^{\frac{\pi }{2}}{x}^2{\cos }^{2n}xdx$$ for $n\ge 0$ and simple estimates.