My attempt:
I used the fact that if $\lim_{n \rightarrow \infty} a_{n} \neq 0 \Rightarrow \sum_{n=1}^{\infty} a_{n}$ doesn´t converges. So, I checked that
\begin{align*} \lim_{n \rightarrow \infty} \left |(-1)^{n}\cos(\frac{\pi}{n}) \right |=1 \end{align*} i.e., \begin{align*} \lim_{n \rightarrow \infty} \left |(-1)^{n}\cos(\frac{\pi}{n}) \right |\neq0 \Longrightarrow \sum_{n=1}^{\infty} \left |(-1)^{n}\cos(\frac{\pi}{n}) \right | \text{does not converges} \end{align*}
My doubt:
Is this implication correct? \begin{align} \sum_{n=1}^{\infty} \left |(-1)^{n}\cos(\frac{\pi}{n}) \right | \text{does not converges} \Longrightarrow \sum_{n=1}^{\infty} (-1)^{n}\cos(\frac{\pi}{n}) \text{ does not converges} \end{align}