Does $\sum_{n=1}^{\infty} \frac{\ln(n)}{n^2+2}$ converge? I'm trying to find out whether
$$\sum_{n=1}^{\infty} \frac{\ln(n)}{n^2+2}$$
is convergent or divergent?

Clearly 
$$
\frac{\ln(n)}{n^2+2}\leq \frac{\ln(n)}{n^2}
$$
You can apply the integral test to show that $\sum\frac{\ln(n)}{n^2}$ converges.  You only need to check that $\frac{\ln(n)}{n^2}$ is decreasing.  But, the derivative is clearly negative for $n>e$.
