Answer
The series converges when $|x|<1$.
Work Step by Step
Given
$$\sum_{n=1}^{\infty} n^2x^{n}$$
By using the Ratio Test, we get:
\begin{align*}
\rho&=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|\\
&=\lim _{n \rightarrow \infty}\left| \frac{(n+1)^2x^{n+1}}{ n^2x^{n}}\right|\\
&= \lim _{n \rightarrow \infty} \frac{ (n+1)^{2}}{n^2}|x|\\
&= |x|
\end{align*}
Thus the series converges when $|x|<1$.
