Answer
The given series converges with a radius of convergence of $|x|\lt 2$.
Work Step by Step
$a_{n}=\sum_{n=0}^{\infty} \frac{b_n}{n+1}x^{n+1}$
$\lim\limits_{n \to \infty}|\frac{a_{n+1}}{a_{n}}|=\lim\limits_{n \to \infty}|\frac{\frac{b_n+1}{(n+1)+1}x^{(n+1)+1}}{\frac{b_n}{n+1}x^{n+1}}|$
$=\lim\limits_{n \to \infty}|\frac{x}{2}|\lt 1$
$=|x|\lt 2$
The given series converges with a radius of convergence of $|x|\lt 2$.
