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I am trying to solve the series

$$\sum_{k=1}^\infty\frac{1}{k(k^2+n^2)}$$

The best I got is

$$\frac{\Re\left\{\psi(1+in) \right\}+\gamma)}{n^2}$$

I am not able to simplify it more.

Maybe there is another approach to solve the series. Any idea how ?

You can assume that $n$ is an integer if that simplifies the solution.

Zaid Alyafeai
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    Notice that for $n=0$ we have $S=\zeta(3),~$ which is notorious for not possessing a known closed form expression, so I'm afraid that no meaningful simplifications to this series exist. – Lucian Jul 17 '16 at 19:26
  • @Lucian, Can't we at least get rid of the complex values ? – Zaid Alyafeai Jul 17 '16 at 19:48

1 Answers1

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In fact, your result is correct and it agrees to Eq.(6.3.17) of Abramovic & Stegun, where you'll find a nice zeta series representation for it. See link http://people.math.sfu.ca/~cbm/aands/page_259.htm

Dr. Fabio M. S. Lima
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