How we can find ROC(Region of convergence) given a signal in z domain?
You need to determine the values of the complex variable $z$ for which the series

$$\sum_{n=1}^{\infty}n^2z^{-n}\tag{1}$$

converges. So you need to know a few things about infinite series.

For this case a simple test is the ratio test. You take the ratio of two successive elements of the series and compute the limit:

$$L=\lim_{n\to\infty}\left|\frac{(n+1)^2z^{-(n+1)}}{n^2z^{-n}}\right|=\lim_{n\to\infty}\left|\frac{(n+1)^2}{n^2}\right|\left|z^{-1}\right|=\frac{1}{|z|}\tag{2}$$

The series converges absolutely for $L&lt;1$, i.e., for $1/|z|&lt;1$, or $|z|&gt;1$. So the ROC is the region $|z|&gt;1$.

Note that the ROC would not change if we used any other power of $n$ in $(1)$.

EDIT:

A step for step explanation of Eq. $(2)$:

$$\begin{align}L&amp;=\lim_{n\to\infty}\left|\frac{(n+1)^2z^{-(n+1)}}{n^2z^{-n}}\right|\\&amp;=\lim_{n\to\infty}\left|\frac{(n+1)^2}{n^2}\right|\left|\frac{z^{-(n+1)}}{z^{-n}}\right|\\&amp;=\lim_{n\to\infty}\left|\frac{(n+1)^2}{n^2}\right|\left|z^{-1}\right|\\&amp;=\frac{1}{|z|}\lim_{n\to\infty}\left|\frac{(n+1)^2}{n^2}\right|\\&amp;=\frac{1}{|z|}\lim_{n\to\infty}\left(\frac{n^2+2n+1}{n^2}\right)\\&amp;=\frac{1}{|z|}\lim_{n\to\infty}\left(1+\frac{2}{n}+\frac{1}{n^2}\right)\\&amp;=\frac{1}{|z|}\left(1+\underbrace{\lim_{n\to\infty}\frac{2}{n}}_{0}+\underbrace{\lim_{n\to\infty}\frac{1}{n^2}}_{0}\right)\\&amp;=\frac{1}{|z|}\end{align}$$