Answer
The given series converges with $R=10$
Work Step by Step
$a_{n}=\sum_{n=1}^{\infty} nc_{n}x^{n-1}$
$\lim\limits_{n \to \infty}|\frac{a_{n+1}}{a_{n}}|=\lim\limits_{n \to \infty}|\frac{n+1c_{n+1}x^{n}}{nc_{n}x^{n-1}}|$
$=\lim\limits_{n \to \infty}\frac{x}{|10|}\lt 1$
$=|x|\lt 10$
The given series converges with $R=10$
