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

Daniel Daners
Preprint (PDF) June 2011
Mathematics Magazine 85 No 5 (2012), 361-364
Original article available at doi:10.4169/math.mag.85.5.361
Citations on Google Scholar


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.

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AMS Subject Classification (2000): 40A25