How can I test whether $\sum (n+\cos(n))/(n^3*\log(n^2+4))$ is convergent or not?

Question

I'm trying to find out whether the series $$ \sum_{n=1}^\infty \frac{n + \cos n}{n^3 \log (n^2+4)} $$

is convergent or not.

I tried the ratio test first, but it seems does work because there will be $\cos(n)$ and $\cos(n+1)$ in numerator and denominator. I can't really find a way to compare those two terms. Then I tried to find the limit of when n goes to infinity but the $\cos(n)$ really confused me. $\cos$ of infinity is a undefined value so I cant really continue on this question. Am I using the right method?

Answer 1

Since$$(\forall n\in\mathbb{N}):\left|\frac{n+\cos n}{n^3\log(n^2+4)}\right|\leqslant\frac{n+1}{n^3}=\frac1{n^2}+\frac1{n^3}\leqslant\frac2{n^2},$$your series converges absolutely.