Showing a numerical sequence converges How could I show that the following sequence converges?
$$\sum_{n = 1}^{\infty} \frac{\sqrt{n} \log n}{n^2 + 3n + 1}$$
I tried the ratio and nth-root tests and both were inconclusive. I was thinking there might be a way to use the limit comparison test, but I'm not sure. Any hints?

Hint:
$$\sum_{n = 1}^{\infty} \frac{\sqrt{n} \log n}{n^2 + 3n + 1} < \sum_{n = 1}^{\infty} \frac{ \log n}{n^{3/2}}$$
Then by integral test, since $\int_{1}^{\infty}\frac{ \log n}{n^{3/2}}=4$ (converges), so the given series converges.
