Difficult integral with dirac comb

Question

I'm trying to solve the following integral:

$\int_{-\infty}^{\infty} \frac{\cos(\pi (x+1))}{x}[\sum_{n=1}^{\infty}\delta (x-n)]dx$

So this seems pretty terrible, and there is also a hint

Hint: "Don't be afraid". Nevertheless, I am afraid.

How do you start solving this? I know I'm supposed to show some effort, but I really have no idea where to start. Maybe integration by parts?

Answer 1

We simply use the "sifting" property of the Dirac Delta to obtain

$$\begin{align} \int_{-\infty}^{\infty}\frac{\cos(\pi(x+1))}{x}\sum_{n=1}^{\infty}\delta(x-n)\,dx&=\sum_{n=1}^{\infty}\frac{\cos(\pi(n+1))}{n}\\\\ &=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}\\\\ &=\log 2 \end{align}$$