TITLE: How to prove $\sum_{n=0}^{\infty} \frac{1}{1+n^2} = \frac{\pi+1}{2}+\frac{\pi}{e^{2\pi}-1}$
QUESTION [28 upvotes]: How can we prove the following

$$\sum_{n=0}^{\infty} \dfrac{1}{1+n^2} = \dfrac{\pi+1}{2}+\dfrac{\pi}{e^{2\pi}-1}$$   

I tried using partial fraction and the famous result $$\sum_{n=0}^{\infty} \dfrac{1}{n^2}=\frac{\pi^2}{6}$$  
But I'm stuck at this problem.

REPLY [2 votes]: Hint: Differentiate the natural logarithm of Euler's infinite product expression for the sine function, then use the well-known relations between trigonometric and hyperbolic functions.