Is the $\sum_{n=1}^{\infty} \frac{(2n+1)^{1/2}}{n^2}$ convergent or divergent? For this question I am not really sure which test to use to determine this. I was thinking the comparison or limit comparison test but it doesn't seem to be working. I was wondering what the steps are to figure this out, and if it is the comparison test, how would you use it? Any help is appreciated!

We have
$$\frac{\sqrt{2n+1}}{n^2}\sim_\infty\sqrt2\frac{1}{n^{3/2}}$$
hence the given series is convergent by the asymptotic comparison with a convergent Riemann series.
