The series is $\sum_{n=0}^{\infty} \frac{2^n x^n}{3^{n+1}}$

The radius of convergence is $\frac{3}{2}$

The series can be written as $\frac{1}{3}\sum_{n=0}^{\infty} \left(\frac{2x}{3}\right)^n$

The series converges for all $x \in \mathbb{R}$