Comparison & Limit Comparison test for series

Can someone see if I am doing this the right way? Or if there is an EASIER way to do this?

Determine whether the series converges or diverges.

$\displaystyle \sum^{\infty}_{n=1} \frac{n^2 + 1}{n^2 \sqrt{n}}$

*Pulled out my variables:

$\displaystyle \sum \frac{n^{\frac{4}{2}}}{n^{\frac{5}{2}}}$

$\displaystyle = \frac{n}{n^{\frac{1}{2}}}$ = $\displaystyle n^{\frac{1}{2}}$

So,

$\displaystyle \lim_{n \rightarrow \infty} \frac{n^{\frac{4}{2}}}{n^{\frac{5}{2}}} \cdot \frac{1}{n^{\frac{1}{2}}} $

$\displaystyle \rightarrow \lim_{n\rightarrow \infty} \frac{n^2}{n^3}$

$\displaystyle \rightarrow \lim_{n\rightarrow \infty} \frac{1}{n} = 0$

BUT, because $\displaystyle n^{\frac{1}{2}}$ is DIVERGENT then this series is also divergent. Right?

This stuff is SO abstract! Any help is GREATLY appreciated! Thanks! Molly