# nth partial sum of the series for $\zeta(2)$

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.

• There isn't a nice closed form. You can cook up integral representations, however. – Chappers Jun 14 '15 at 21:31
• The standard notation is $$H_n^{(2)}=\sum_{k=1}^{n}\frac{1}{k^2}.$$ – Jack D'Aurizio Jun 14 '15 at 21:35
• This was discussed by several users at this MSE link. – Marko Riedel Jun 14 '15 at 21:35
• 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. – alex.jordan Jun 14 '15 at 21:41
• My previous comment is just smoke and mirrors though, since ${n+1\brack 2}$ is not any easier to compute than $H_n^{(2)}$. – alex.jordan Jun 14 '15 at 21:48