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}
Best Answer
Since $ \lim\limits_{n\to +\infty}{\left(-1\right)^{n}\cos{\left(\frac{\pi}{n}\right)}} $ is not $ 0 $, then your series diverges.