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 (\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.