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How to evaluate $\displaystyle\sum_{k=1}^\infty\left(\frac{\sec{(k\pi\sqrt{5})}}{k^2}\right)^2$?

In general, how to find the closed form of infinite series $\displaystyle\sum_{k=1}^\infty\left(\frac{\sec{(kz)}}{k^2}\right)^2$ in term of variable $z$ ? Thanks in advance.

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  • $\begingroup$ what makes u believe that there is aclosed form? $\endgroup$ – tired Jan 10 '16 at 15:46
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    $\begingroup$ Are you sure it converges? It probably does, but this is likely not a trivial result (and related to the irrationality measure of $\pi$). $\endgroup$ – Winther Jan 10 '16 at 15:53
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    $\begingroup$ Disregard the last note in the comment above, I misread the sum. It has nothing to do with the irrationality measure of $\pi$ when $z = \pi \sqrt{5}$, but rather with that of $\sqrt{5}$. $\endgroup$ – Winther Jan 10 '16 at 16:12
  • $\begingroup$ From wolfram alpha, the series is seem like converge. However, I can't prove it. I don't know is there a closed from. I'm just looking for possibilities. $\endgroup$ – Zoll Jan 10 '16 at 16:21

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