Chapter 11 - Section 11.9 - Representations of Functions as Power Series - 11.9 Exercises - Page 757: 1

The given series converges with $R=10$

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