What is the sum of the series $\sum\limits _{k=1}^\infty \frac{1}{k^2}$?

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

What is $\lim \limits_{n\to\infty} \sum\limits_{k=1}^n \frac{1}{k^2}$ as an exact value?

http://en.wikipedia.org/wiki/Riemann_zeta_function#Specific_values — JavaMan, Apr 20 '11 at 15:57
I suggest you to google 'Basel's problem'. Then you will get a numerous resource on it, including diverse proofs. — Sangchul Lee, Apr 20 '11 at 16:03
Don't miss Robin Chapman's collection of 14 proofs here: http://empslocal.ex.ac.uk/people/staff/rjchapma/etc/zeta2.pdf — , Apr 20 '11 at 16:12
Just because it's Xmas doesn't mean you can resurrect closed questions with pointless edits, Rahul! — The Chaz 2.0, Dec 25 '11 at 05:56

score 1 · Answer 1 · answered Apr 20 '11 at 16:23

This is a very well-known series. Astoundingly enough, the exact value is $\dfrac{\pi^2}{6}$.

I prefer the last one, as it's closely related to Viete's infinite product for $\pi$.

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