Question

sum of cos^2/n^2+1 from 1 to infinity, converges or diverges?

Answer 1

$-1\le \cos x\le1\implies 0\le \cos^2x\le1$
$\implies\displaystyle\sum_{n=1}^\infty \frac{\cos^2n}{n^2+1}\le\sum_{n=1}^\infty \frac1{n^2+1}<\sum_{n=1}^\infty\frac1{n^2}$

so the series converges by comparison, as $\sum\frac1{n^2}$ is known to converge.