Chapter 11 - Infinite Sequences and Series - 11.8 Power Series - 11.8 Exercises - Page 791: 3

$R=1$ ; interval of convergence is $(-1,1)$

Let $a_{n}=(-1)^{n}nx^{n}$, then $\lim\limits_{n \to \infty}|\frac{a_{n+1}}{a_{n}}|=\lim\limits_{n \to \infty}|\frac{(-1)^{n+1}(n+1)x^{n+1}}{(-1)^{n}nx^{n}}|$ $=|x|$ converges if $|x|\lt 1$, so $R=1$ and the interval of convergence is from $-1$ to $+1$. Hence, $R=1$ ; interval of convergence is $(-1,1)$