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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    $\begingroup$ There's a nice discussion at math.stackexchange.com/questions/1322086/… $\endgroup$ – Gerry Myerson Apr 12 at 12:15
  • $\begingroup$ Anyway the question is not solved at that MathSE link. Yet it would be useful to mention what it says. $\endgroup$ – YCor Apr 13 at 11:06
  • $\begingroup$ 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-Puchta Apr 26 at 9:23

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.


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