Absolute convergence of series with cosines

Question

Consider the series: \begin{equation} \sum\limits _{n=1}^{+\infty}\cos\left(nx\right)\cos\left(ny\right),\quad x,y \in \left(0,\pi\right) \end{equation} What is the easiest way to prove that the series converges absolutely for $x\neq y$?

Answer 1

It does not converge absolutely, since for $x=\dfrac{\pi}{4}$ and $y=\dfrac{3\pi}{4}$ and $n=4k$ we get:

$|cos(k\pi)||cos3k\pi|=1.1=1$ . Summing over all $n=4k, k=1,2,3,...$

we get that the series diverges!