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I have the summation

$$\sum\limits_{i=1}^n \frac{1}{i^2}$$

And I don't know how to find a closed formula for it. Any ideas?

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  • $\begingroup$ It's actually a know problem. $\endgroup$ – Piotr Benedysiuk Jul 3 '16 at 20:57
  • $\begingroup$ That is the generalized harmonic number $H_n^{(2)}$. $\endgroup$ – Jack D'Aurizio Jul 3 '16 at 21:05
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    $\begingroup$ @PiotrBenedysiuk Basel's problem is the evaluation of $\sum\limits_{k=1}^\infty k^{-2}$ while the question here is a closed formula for $g(n)=\sum\limits_{k=1}^n k^{-2}$. Ok, arguably one may say that, if we had a closed form for $g(n)$, then $\lim_{n\to\infty}g(n)$ would not be a "problem", but that link does not talk exactly of this problem. $\endgroup$ – user228113 Jul 3 '16 at 21:26
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The infinite series $$\sum_{i=1}^\infty \frac{1}{i^2}$$ has a closed form. But the finite series does not.

Sometimes we may write it using the "trigamma" function, but this is not much beyond the definition. $$ \sum_{i=1}^n \frac{1}{i^2} = \frac{\pi^2}{6} - \psi^{(1)}(n+1) $$

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    $\begingroup$ +1. I'll take $\underline{this\ one}$ as a closed expression. $\endgroup$ – Felix Marin Jul 4 '16 at 1:51

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