Is there a general method or formula for calculating the infinite sum $\sum_{n=1}^{\infty} 1/(an^{2}+bn+c) $?
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2$\begingroup$ There's a nice discussion at math.stackexchange.com/questions/1322086/… $\endgroup$– Gerry MyersonCommented Apr 12, 2019 at 12:15
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$\begingroup$ Anyway the question is not solved at that MathSE link. Yet it would be useful to mention what it says. $\endgroup$– YCorCommented Apr 13, 2019 at 11:06
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$\begingroup$ The Stackeschange answer does not solve the problem, and it is so non-trivial that it could well be asked on MathOverflow. $\endgroup$– Jan-Christoph Schlage-PuchtaCommented Apr 26, 2019 at 9:23
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1 Answer
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It depends what you call "calculating". If you want the result in "closed form", any series of rational functions of $n$ can be summed in terms of the logarithmic derivative $\psi$ of the gamma function and its derivatives, simply by decomposing into partial fractions. If you want a numerical approximation, you can either use this, or the many summation methods existing in the literature which can give you thousands of decimals in fractions of a second.
answered Apr 12, 2019 at 12:17