Compute the Z-transform of a^{-n} and step function

[asd1249jf]

Homework Statement

Compute the Z-transform of

$$x[n] = (0.5)^{-n} * u[n-1]$$

And find the ROC (Region of Convergence)

Homework Equations

Z-Transform for discrete time
$$x(z) = \sum_{n=0}^\infty x[n] * z^{-n}$$

The Attempt at a Solution

To solve this, I used the definition of Z-transform

$$x(z) = \sum_{n=1}^\infty (0.5)^{-n} * z^{-n}$$

Note that the summation starts from n = 1 due to the time shift to the right by 1.

Simplifying this, we get

$$x(z) = \sum_{n=1}^\infty (0.5z)^{-n}$$

Here's where I'm confused as hell. Apparently, we can't apply the geometric series to simplify this further since the expression is powered to the negative n.

Furthermore, what's contradicting about this problem is that the series diverges. Does this indicate that the z-transform does not exist at all?

Any help will be appriciated.

 

[wildman]

Try a little rewrite:

$$x(z) = \sum_{n=1}^\infty (0.5^{-1}z^{-1})^{n}$$ or

$$x(z) = \sum_{n=1}^\infty (2z^{-1})^{n}$$ Now use your geometric series...