I want to derive the formula for $1/(1^2)+1/(2^2)+...+1/(n^2)$ where $n$ is a positive integer and $n<\infty$. All I could find no the internet dealt with the case where $n$ goes to infinity, and based on those solutions, I'm not even sure that a formula exists for the finite case. I tried deriving it by induction, but failed. Any help will be appreciated.
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PiMan
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2Unlikely there is a closed form. As with the Harmonic numbers, there should be useful analytic approximations. See, e.g. this – lulu May 01 '20 at 11:27
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1A first approximation is $\frac{\pi^2}{6}-\frac1n$ – saulspatz May 01 '20 at 11:30