Example of a z-transform
let $ x[n]=\frac{u[n]}{2^n} $
$ X(z) = \sum_{n=-\infty}^{\infty}x[n]z^{-n} $
$ = \sum_{n=-\infty}^{\infty}\frac{u[n]}{2^n}z^{-n} $
$ = \sum_{n=0}^{\infty}\frac{1}{2^n}z^{-n} $
$ =\sum_{n=0}^{\infty}{\frac{1}{2z}}^n $
if $ |\frac{1}{2z}| >= 1 $ then X(z) diverges
else
$ X(z)=\frac{1}{1-\frac{1}{2z}} = \frac{2z}{2z-1} $
therefore the Z.T is $ \frac{2z}{2z-1} $ ROC: $ |\frac{1}{2z}|<1 $ or $ \frac{1}{2}<|z| $
