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Different methods to compute $\sum_{n=1}^\infty \frac{1}{n^2}$.

$\sum \frac{1}{n^2}= \frac{\pi^2}{6}$

There are a few proofs for that fact but can anybody see why is it "really" true? Talking in geometric terms for example.

  • 2
    $\begingroup$ There are some relatively geometric proofs here: math.stackexchange.com/questions/8337/… $\endgroup$ – Qiaochu Yuan May 20 '11 at 8:33
  • $\begingroup$ I am voting to close. $\endgroup$ – user9413 May 20 '11 at 8:45
  • $\begingroup$ When you want to know that it is "really" true you usally look at the analytic proof not the geometric interpretation. $\endgroup$ – Listing May 20 '11 at 10:21
  • $\begingroup$ @Qiaouchu Yuan: Thank you. I didn't know this question was suggested already. $\endgroup$ – Tau May 20 '11 at 10:22
  • $\begingroup$ @Qiaochu: Even though there are many proofs, there should be room for an intuitive explanation, a 'why' answer is more than just 'how'. Would a rewording of the question to that effect be a question not likely to be closed? Naively $\pi$ has something to do with circles, and the summation not at all, and that warrants a human explanation in addition to the algebraic directions for the path between them. $\endgroup$ – Mitch May 20 '11 at 14:27