Answer
Converges
Work Step by Step
Let $u_n=\dfrac{1+\cos n}{n^2}$
But $ 0\leq \cos^2 n \leq 1$
This implies that $u_n \leq \dfrac{2}{n^2}$
Here, $\Sigma_{n=1}^\infty \dfrac{2}{n^2}$ shows a convergent p-series with $p=2$
Hence, the series converges due to the Direct comparison test.