Answer
$R=1$ ; interval of convergence is $(-1,1)$
Work Step by Step
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 $(-1,1)$
