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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?


marked as duplicate by David Mitra, Henry, Asaf Karagila, Martin Sleziak, Nate Eldredge May 13 '12 at 20:35

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    $\begingroup$ That should be $\pi^2/6$. $\endgroup$ – David Mitra May 13 '12 at 19:31
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    $\begingroup$ See here $\endgroup$ – David Mitra May 13 '12 at 19:33
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    $\begingroup$ @Martin: This question asks for methods to calculate the sum, not to prove its convergence. This is at least how I read this question. $\endgroup$ – Asaf Karagila May 13 '12 at 19:55
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    $\begingroup$ @Asaf I think you' right. Although, the body is different from the title. Which explains why I thought that the OP asks about convergence only. (It's not that important now, since we found duplicates for both possible meanings.) $\endgroup$ – Martin Sleziak May 13 '12 at 19:57
  • $\begingroup$ The title originally said Prove that this series converges $\endgroup$ – Henry May 14 '12 at 10:05

Fourteen proofs compiled by Robin Chapman.



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 $\int_{k}^{k+1} \frac{1}{x^2} dx \lt \frac{1}{k^2} \lt \int_{k-1}^{k} \frac{1}{x^2} dx$.


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