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Let $$F(x) = \sum _{n=1}^{\infty} \frac{1}{n^2 - x^2}$$

It seems that for odd integer $k$ $$F\left(\frac{k}{2}\right) = \frac{2}{k^2}$$ My evidence is strictly computational, and I have no idea how to approach a proper proof. So standard questions are:

  • is it a known fact
  • is it a fact
  • is it curious
  • what is a proof strategy

Please advise.

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marked as duplicate by Dominic Michaelis, Norbert, Dennis Gulko, vonbrand, Davide Giraudo Mar 28 '13 at 11:00

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

3… and put $a= ix$ – Cortizol Mar 28 '13 at 8:39
Or more directly the question 141470 – Raymond Manzoni Mar 28 '13 at 8:41
Thanks everyone. Sorry for being illiterate. – user58697 Mar 28 '13 at 9:00