I could just prove that the sequence is bounded but couldn't find the exact limit.
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1You just need the existence. BTW, the exact limit is $\dfrac{\pi ^2}{6}$. – FormerMath Nov 19 '14 at 13:27
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But how to prove that the limit is nearly 2 – E and pi Nov 19 '14 at 13:28
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Here: http://en.m.wikipedia.org/wiki/Basel_problem – FormerMath Nov 19 '14 at 13:33
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See also Robin Chapman's notes about this problem here: http://empslocal.ex.ac.uk/people/staff/rjchapma/etc/zeta2.pdf – Travis Willse Nov 19 '14 at 13:37
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Please state exactly what you are trying to do: (a) prove the existence of limit; (b) show it's "nearly 2" (?); (c) find its exact value. – Nov 19 '14 at 14:17
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The series is (obviously) $\sum\limits_{i=1}^\infty \frac{1}{n^2}$. And thus $\lim_{n \to \infty} \sum\limits_{i=1}^\infty \frac{1}{n^2} = \frac{pi^2}{6}$. See also the hyperharmonic/p-series for more information. This problem is called Basel problem. See wikipedia for a detailed proof.
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