What is a closed form formula for the following?$$\sum_{r=1}^n\frac{1}{r^2}$$ I know that the summation $$\sum_{r=1}^\infty\frac{1}{r^2}=\frac{\pi^2}{6}$$ but is there a formula for the finite series? Thank you.
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1Does this answer your question? Do harmonic numbers have a “closed-form” expression? – Kyan Cheung Oct 23 '20 at 12:55
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The finite sum is $\frac{{\pi ^2 }}{6} + \frac{1}{{n^2 }} - \zeta (2,n)$, where $\zeta(s,a)$ is the Hurwitz zeta function. You can obtain a number of different representations for your sum via known representations of the Hurwitz zeta function, see, e.g., https://dlmf.nist.gov/25.11 – Gary Oct 23 '20 at 12:59