[tex]-1\le \cos x\le1\implies 0\le \cos^2x\le1[/tex]
[tex]\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}[/tex]
so the series converges by comparison, as [tex]\sum\frac1{n^2}[/tex] is known to converge.
