Please, can you advice me, how to determine the $n$th partial sum of the finite series $1 + 1/2^2 + 1/3^2 + \dots + 1/n^2$? Thank you very much.

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    $\begingroup$ There isn't a nice closed form. You can cook up integral representations, however. $\endgroup$ – Chappers Jun 14 '15 at 21:31
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    $\begingroup$ The standard notation is $$ H_n^{(2)}=\sum_{k=1}^{n}\frac{1}{k^2}.$$ $\endgroup$ – Jack D'Aurizio Jun 14 '15 at 21:35
  • $\begingroup$ This was discussed by several users at this MSE link. $\endgroup$ – Marko Riedel Jun 14 '15 at 21:35
  • $\begingroup$ This OEIS entry asserts that $\sum_{k=1}^n\frac{1}{n^2}=\frac{1}{(n!)^2}\left({n+1\brack 2}^2-2{n+1\brack1}{n+1\brack3}\right)$, where $a\brack b$ is a Stirling number of the first kind. That's sort of a closed formula if you have a way to compute $a\brack b$ easily. $\endgroup$ – alex.jordan Jun 14 '15 at 21:41
  • $\begingroup$ My previous comment is just smoke and mirrors though, since ${n+1\brack 2}$ is not any easier to compute than $H_n^{(2)}$. $\endgroup$ – alex.jordan Jun 14 '15 at 21:48

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