Answer
Radius of convergence is: $\dfrac{2}{3}$
Work Step by Step
We need to apply the Ratio Test to the series.
$\lim\limits_{n \to \infty} |\dfrac{u_{n+1}}{u_n}|=\lim\limits_{n \to \infty} |\dfrac{(3n+1)x}{2n+2}|$
or, $=|x| \lim\limits_{n \to \infty} (\dfrac{(3n+2)}{2n+2})$
or, $=|x| \lim\limits_{n \to \infty} (\dfrac{(3+2/n)}{n+2/n})$
So, $=\dfrac{3}{2}|x|$
The series converges absolutely for $\dfrac{3}{2}|x| \lt 1$ or, $|x| \lt \dfrac{2}{3}$
So, the radius of convergence is: $\dfrac{2}{3}$
