Answer
The series $\sum_{n=1}^{\infty} \frac{1}{3^{n^2}} $ converges.
Work Step by Step
We compare the given series
$\sum_{n=1}^{\infty} \frac{1}{3^{n^2}} $
with the geometric series
$\sum_{n=1}^{\infty} \frac{1}{3^{n}} $
which is a convergent series ($r=\frac{1}{3}\lt 1$).
Thus, by the comparison test, the given series $\sum_{n=1}^{\infty} \frac{1}{3^{n^2}} $ also converges.
