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Possible Duplicate:
Different methods to compute $\sum\limits_{n=1}^\infty \frac{1}{n^2}$
Does $\sum\limits_{k=1}^n 1 / k ^ 2$ converge when $n\rightarrow\infty$?

I read my book of EDP, and there appears the next serie $\sum _{k=1} \dfrac{1}{k^2} = \dfrac{\pi^2}{6}$ And, also, we prove that this series is equal $\frac{\pi^2}{6}$ for methods od analysis of Fourier, but...

Do you know other proof, any more simple or beautiful?

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    The title ori$g$inally said **Prove that this series conver$g$es**2012-05-14

2 Answers 2

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Fourteen proofs compiled by Robin Chapman.

http://empslocal.ex.ac.uk/people/staff/rjchapma/etc/zeta2.pdf

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If you just want to show it converges, then the partial sums are increasing but the whole series is bounded above by $1+\int_1^\infty \frac{1}{x^2} dx=2$ and below by $\int_1^\infty \frac{1}{x^2} dx=1,$ since $\int_{k}^{k+1} \frac{1}{x^2} dx \lt \frac{1}{k^2} \lt \int_{k-1}^{k} \frac{1}{x^2} dx$.