Answer
The radius of convergence of the series is $R=\dfrac{1}{3}$.
Work Step by Step
Here, we have $a_n=(3x)^n$
Root Test states that when $\Sigma a_n$ is an infinite series with positive terms and, then $r=\lim\limits_{n \to \infty}\sqrt[n] a_n$
a) When $0 \leq r \lt 1$, the series converges. (b) When $r \gt 1$, or, $\infty$, so the series diverges. (c) When $r=1$, the ratio test is inconclusive.
Now, $r=\lim\limits_{n \to \infty}\sqrt[n] {(3x)^n}\\=|3x|$
Therefore, the series converges for $|3x| \lt 1$ or, $|x| \lt \dfrac{1}{3}$ by the root test. This implies that the radius of convergence of the series is $R=\dfrac{1}{3}$.
