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I was reading sequence and series and there I saw the formula for the sum of the squares of the natural numbers ($1^2+2^2+3^2+\cdots+n^2$) so I just incurred a doubt about the sum obtained when these squared numbers are inversed i.e $$\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2}$$ I tried a lot but did not reach a conclusion. I tried to convert the whole sum of the components into a single fraction, tried to take commons etc. but did not succeed. So is there a way to find a general form for the sum ? If yes please help me out. Any help is appreciated. $$Thank you$$

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    $\begingroup$ This is a generalized harmonic number. I don't think there is a known closed form for the finite sum, but it's well known that in the limit, $\sum_{n=1}^{\infty}\frac{1}{n^2} = \pi^2/6$ $\endgroup$
    – user169852
    Commented Dec 13, 2016 at 18:20

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If you take common denominator it should be this $$\frac{\sum_i \prod_{j\neq i} j^2}{\prod_i i^2}$$ It is not the simplest possible BUT it is workable. Albeit not terribly pretty.

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