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Evaluate the sum of the following series

$$\sum_{n=1}^\infty\frac{1}{n^2+4}$$

I saw a video in youtube where it is solved using complex analysis. What other method can be used to solve this?

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    $\begingroup$ Hi Kalpan; Can you post the link? $\endgroup$ – bobbym Nov 3 '14 at 8:47
  • $\begingroup$ One can use the Poisson summation formula. $\endgroup$ – Start wearing purple Nov 3 '14 at 8:49
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    $\begingroup$ youtube.com/watch?v=99Tnu0-0S9g $\endgroup$ – Kalpan Nov 3 '14 at 8:49
  • $\begingroup$ Hi @O.L.Thanks! The problem is solved in the video more or less similar to your answer. In fact , it is done almost the same way as in the 2nd answer in the link you mentioned. $\endgroup$ – Kalpan Nov 3 '14 at 9:07
  • $\begingroup$ Have you tried to use fourier analysis ? One can calculate the sum of the $\frac{1}{n^2}$ using a suitable function, here the term is different but I think it's worth looking into it.. $\endgroup$ – mvggz Nov 3 '14 at 9:12
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After asking the same question to my prof, he get's this:

$$\sum_{n=1}^\infty\frac{1}{n^2+4}=$$ $$\frac{1}{8}\left(2\pi coth(2\pi )-1\right)=$$ $$\frac{1}{8}\left(2\pi \left(1+\frac{2}{-1+e^{4\pi}}\right)-1\right)=$$ $$\frac{1}{8}\left(2\pi +\frac{4\pi}{e^{4\pi}-1}-1\right)=$$ $$\frac{1+2\pi +2\pi e^{4\pi}-e^{4\pi}}{8e^{4\pi}-8}$$

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