Chapter 11 - Section 11.9 - Representations of Functions as Power Series - 11.9 Exercises - Page 757: 20

$R=1$ $\sum_{n=1}^{\infty}n^{2}x^{n}$

$f(x)=\frac{x^{2}+x}{(1-x)^{3}}=\sum_{n=1}^{\infty}n^{2}x^{n}$ $\lim\limits_{n \to \infty}|\frac{a_{n+1}}{a_{n}}|=\lim\limits_{n \to \infty}|\frac{(n+1)^{2}x^{n+1}}{n^{2}x^{n}}|$ $=|x|\lt 1$ The given series converges with $R=1$