Answer
Divergent
Work Step by Step
In the given problem:
$a_{n}=\frac{(-5)^{n}}{n^{2}9^{n}}=\frac{(25)^{n}}{n^{2}9^{n}}$
and
$a_{n+1}=\frac{(25)^{n+1}}{(n+1)^{2}9^{n+1}}$
Re-write $a_{n+1}$ as
$a_{n+1}=\frac{(25)^{n+1}}{(n+1)^{2}9^{n+1}}$
Therefore,
$\frac{a_{n+1}}{a_{n}}= \frac{\frac{(25)^{n+1}}{(n+1)^{2}9^{n+1}}}{\frac{(25)^{n}}{(n)^{2}9^{n}}}$
$=\frac{25}{9}(\frac{n}{n+1})^{2}$
and
$L=\lim\limits_{n \to \infty}|\frac{25}{9}(\frac{n}{n+1})^{2}|$
$=\frac{25}{9} \gt 1$
Thus, $L\gt 1$
Hence, the series is divergent by ratio test.