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

I just got the "New and Revised" edition of "Mathematics: The New Golden Age", by Keith Devlin. On p. 64 it says the sum is $\pi^2/6$, but that's way off. $\pi^2/6 \approx 1.64493406685$ whereas the sum in question is $\approx 1.29128599706$. I'm expecting the sum to be something interesting, but I've forgotten how to do these things.

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    Devlin is right, and 1.29128599706 is incorrect. Since many proofs of this are given at http://math.stackexchange.com/questions/8337/different-methods-to-compute-sum-n-1-infty-frac1n2, I think this should be closed as duplicate.2010-12-16
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    There are plenty of ingenious proofs of this you should read the posts of http://math.stackexchange.com/questions/8337/different-methods-to-compute-sum-n-1-infty-frac1n2 and check the links.2010-12-16
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    Could someone please fix the LaTeX of the question/title?2010-12-16
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    Oh, I see, your incorrect approximation is the approximate value of $$\sum_{n=1}^\infty \frac{1}{n^n}.$$ http://oeis.org/A0730092010-12-16

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The answer is indeed pretty interesting!

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

This can be proven using complex analysis or calculus, or probably in many hundreds of other ways. One example of how to prove this is given here:

http://www.math.uu.se/~bjorklund/euler.pdf