Answer
Diverges.
Work Step by Step
We have the given series
$\sum_{n=1}^{\infty} \frac{\sqrt{n}}{4 n+9}$
We apply the limit comparison test with $b_n=(\frac{1}{n^{1/2}})$ (a divergent p-series with $p=1/2\lt 1$):
$L=\lim_{n\rightarrow\infty} \frac{a_n}{b_n}=\lim_{n\rightarrow\infty}\frac{\sqrt{n}}{4 n+9}\times n^{1/2}=\lim_{n\rightarrow\infty}\frac{n}{4n+9}=1/4$
Since $L=1/4\gt 0$, our starting series diverges.
