Does this series converge absolutely: $\sum\limits_{n=1}^{\infty} \frac{\cos n}{n}$?

Question

$$\sum_{n=1}^{\infty} \frac{\cos n}{n}$$ I know that this series converges (from Dirichlet test) but I can't figure if it converges absolutely or not, obviously ratio and root test won't work, limit comparison test seems to fail as well.

Answer 1

It is not absolutely convergent.

Since $|\cos n|\leq 1$, we have $\cos^2n \leq |\cos n|$, so it suffices to show that $\sum_{n=1}^\infty \dfrac{\cos^2n}{n}$ diverges. Now $$\dfrac{\cos^2n}{n}=\dfrac{1+\cos2n}{2n}=\dfrac{1}{2n}+\dfrac{\cos 2n}{2n},$$ and $\sum_{n=1}^\infty \dfrac{1}{2n}$ diverges (harmonic series), so it suffices to show that $\sum_{n=1}^\infty \dfrac{\cos2n}{2n}$ converges. But as you know this follows from the Dirichlet Test.