The above series converges if $\vert z\vert>1$ .
One should note that the Unit step sequence is defined as

$\displaystyle u_s(k)$	$\displaystyle = 1, \;\;$ for $\displaystyle \;\; k= 0,1,2 \cdots$
	$\displaystyle =0, \;\;$ for $\displaystyle \;\; k < 0$

with a same Z-transform.

Unit ramp function is defined as:

$\displaystyle u_r(t)$	$\displaystyle = t, \;\;$ for $\displaystyle \;\; t \ge 0$
	$\displaystyle =0, \;\;$ for $\displaystyle \;\; t < 0$

The Z-transform is:

$\displaystyle U_r(z)$	$\displaystyle =$	$\displaystyle \frac {Tz}{(z-1)^2}$
	$\displaystyle =$	$\displaystyle T \frac {z^{-1}}{(1-z^{-1})^2}$

with ROC $\vert z\vert>1$ .
For a polynomial function

, the Z-transform is:

$\displaystyle X(z)$	$\displaystyle =$	$\displaystyle \frac{1}{1-a.z^{-1}}$
	$\displaystyle =$	$\displaystyle \frac {z}{z-a}$

where ROC is $\vert z\vert>a$ .