Answer
The series converges.
Work Step by Step
We have the given series
$\sum_{n=4}^{\infty} \frac{1}{n^{2}-9}$
We apply the limit comparison test with $b_n=(\frac{1}{n^2})$ (a convergent p-series with $p=2\gt 1$):
$L=\lim_{n\rightarrow\infty} \frac{1}{n^{2}-9}\times n^2=\lim_{n\rightarrow\infty}\frac{n^2}{n^{2}-9}=1$
Since $L=1$, our starting series converges.
