I'm having a hard time doing this problem. As this is homework, I'd appreciate a guidance towards a solution, not a full answer. Thanks in advance.
I need to prove that the series
$$\sum_{k=1}^\infty \frac{3k}{k^2+k} \sqrt\frac{\ln(k)}{k}$$converges. I've found that since
$$\frac{3k}{k^2+k} \sqrt\frac{\ln(k)}{k} \leq \frac{3}{k}\sqrt\frac{\ln(k)}{k}=3\sqrt\frac{\ln(k)}{k^3}$$
we see that
$$\sum_{k=1}^\infty \frac{3k}{k^2+k} \sqrt\frac{\ln(k)}{k}\leq \sum_{k=1}^\infty 3\sqrt\frac{\ln(k)}{k^3}$$
this is the best approximation I've managed to do, while the series still being convergent, but can't find a way to prove it. I'd love some feedback on whether my approximation is prudent and/or how to proceed towards a solution.