Confused about region of convergence in $z$ transform.

Question

If $X_1(z)=\dfrac{z}{z-2}$ and $X_2(z)=\dfrac{z}{z+2},$ then what will be the difference between ROCs of $X_1$ and $X_2?$

Answer 1

So, we have \begin{align*} X_1(z)&=\frac{1}{1-2z^{-1}}\\ X_2(z)&=\frac{1}{1+2z^{-1}}. \end{align*} From the wiki article, we can see that the ROC for $X_1$ is $|z|>|2|,$ or $|z|>2.$ But we can rewrite $X_2$ as $$X_2(z)=\frac{1}{1-(-2)z^{-1}}, $$ which we can see will have an identical ROC.