Answer
The interval of convergence is $(-1,1)$.
Work Step by Step
We apply the ratio test
$$
\rho=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=\lim _{n \rightarrow \infty} \frac{(n+1)x^{n+1} }{nx^n}=|x|\lim _{n \rightarrow \infty} \frac{n+1}{n}=|x|
$$
Hence, the series $\Sigma_{n=0}^{\infty} nx^n$ converges if and only if $|x|\lt1$. That is, the interval of convergence is $(-1,1)$.
