Answer
The series $\Sigma_{n=1}^{\infty}a_n^2$ converges.
Work Step by Step
Let $b_n=a_n^2$; then applying the ratio test, we have:
$$
\rho=\lim _{n \rightarrow \infty}\left|\frac{b_{n+1}}{b_{n}}\right|=\lim _{n \rightarrow \infty} \left|\frac{a_{n+1}^2}{a_{n}^2}\right|=\frac{1}{9}\lt1
$$
Hence, the series $\Sigma_{n=1}^{\infty}a_n^2$ converges.
