How to prove this cosine series is not absolutely convergent?

Question

I think the series $\sum_{k=0}^{\infty}\vert\cos(ak)\vert$, where $a$ is a non-zero constant, is absolutely convergent. But I had a trouble proving it. Can anyone give me a hint on how to prove it? Thanks!

Answer 1

This is not true. For instance, look at what happens when $a=\pi/2$. We have $$\limsup_n \vert \cos\left(n\pi/2 \right)\vert = 1$$

In fact this is not true for any $a$, since we can prove that for any $a$, we have $$\limsup_n \vert \cos\left(na \right)\vert = 1$$