Editing Z-transform (section)

===Example 2 (causal ROC)===
[[Image:Region of convergence 0.5 causal.svg|thumb|250px|ROC (blue), {{pipe}}''z''{{pipe}}&nbsp;=&nbsp;.5 (dashed black circle), and the unit circle (dotted grey circle).]]

Let <math>x[n] = (.5)^n \, u[n] </math> (where <math>u</math> is the [[Heaviside step function]]). Expanding <math>x[n]</math> on the interval <math>(-\infty, \infty)</math> it becomes

:<math>x[n] = \left \{\dots, 0, 0, 0, 1, (.5), (.5)^2, (.5)^3, \dots \right \}.</math>

Looking at the sum

:<math>\sum_{n=-\infty}^{\infty}x[n]z^{-n} = \sum_{n=0}^{\infty}(.5)^nz^{-n} = \sum_{n=0}^{\infty}\left(\frac{.5}{z}\right)^n = \frac{1}{1 - (.5)z^{-1}}.</math>

The last equality arises from the infinite [[geometric series]] and the equality only holds if <math>|(.5)z^{-1}| < 1 ,</math> which can be rewritten in terms of <math>z</math> as <math>|z| > (.5).</math> Thus, the ROC is <math>|z| > (.5).</math> In this case the ROC is the complex plane with a disc of radius 0.5 at the origin "punched out".{{clear}}