Determine whether the series $\sum_{n=1}^{\infty} \frac{\ln(n)}{n^2 +1}$ converges or not.

Question

Determine whether the series $\sum_{n=1}^{\infty} \frac{\ln(n)}{n^2 +1}$ converges or not.

** My trial **
I tried dividing $\frac{\ln(n)}{n^2 +1}$ by $1/n^2$ and finding the limit which was $\infty$ so I could not use the limit comparison test and this idea did not work.

Could anyone give me a hint for studying the convergence of this series?

Answer 1

Compare with $$\sum_{n=1}^{\infty}\frac{1}{n^{1.5}}$$