Question: What is the value of $$\sum\_{n=1}^{\infty}\frac1{1+n^2}$$
I don't know how to solve it.  

I tried to solve it by power series, but I failed.

Answer: 
Hint:
$$\frac{\pi x}{2}\coth \pi x=\frac{1}{2}+\sum\_{n=1}^{\infty }\frac{1}{\left ( \frac{n}{x} \right )^2+1}$$
at $x=1$
$$\sum\_{n=1}^{\infty }\frac{1}{n^2+1}=\frac{\pi}{2}\coth \pi -\frac{1}{2}$$
**Hint.** It's [well-known](https://en.wikipedia.org/wiki/Trigonometric_functions#Cotangent) that
$$\pi\cot(\pi z)=\frac{1}{z}+2\sum\_{n=1}^\infty \frac{z}{z^2-n^2}$$