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

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

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AMS Subject Classification (2000): 40A25