Summation of the series$\sum_{n=1}^\infty\frac{1}{n^2+4}$ 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?

After asking the same question to my prof, he get's this:
$$\sum_{n=1}^\infty\frac{1}{n^2+4}=$$
$$\frac{1}{8}\left(2\pi coth(2\pi )-1\right)=$$
$$\frac{1}{8}\left(2\pi \left(1+\frac{2}{-1+e^{4\pi}}\right)-1\right)=$$
$$\frac{1}{8}\left(2\pi +\frac{4\pi}{e^{4\pi}-1}-1\right)=$$
$$\frac{1+2\pi +2\pi e^{4\pi}-e^{4\pi}}{8e^{4\pi}-8}$$
