Answer
The radius of convergence is equal to: $\dfrac{2}{3}$
Work Step by Step
Apply the Ratio Test to the series as follows:
$$\lim\limits_{n \to \infty} |\dfrac{u_{n+1}}{u_n}|=\lim\limits_{n \to \infty} |\dfrac{(3n+1)x}{2n+2}| \\=|x| \times \lim\limits_{n \to \infty} (\dfrac{(3n+2)}{2n+2}) \\=|x| \times \lim\limits_{n \to \infty} (\dfrac{(3+2/n)}{n+2/n})\\=\dfrac{3}{2}|x|$$
But the series is absolutely convergent for $\dfrac{3}{2}|x| \lt 1 \implies |x| \lt \dfrac{2}{3}$
Thus, the radius of convergence is equal to: $\dfrac{2}{3}$
