Chapter 11 - Section 11.9 - Representations of Functions as Power Series - 11.9 Exercises - Page 757: 19

$\sum_{n=0}^{\infty}(2n+1)x^{n}$, $R=1$

$f(x)=\frac{1+x}{(1-x)^{2}}=\sum_{n=0}^{\infty}(2n+1)x^{n}$ $\lim\limits_{n \to \infty}|\frac{a_{n+1}}{a_{n}}|=\lim\limits_{n \to \infty}|\frac{(2(n+1)+1)x^{n+1}}{(2n+1)x^{n}}|$ $=|x|\lt 1$ The given series converges with $R=1$