Z Transform of n-varying function

Question

I've been doing some reading on z-transforms and I'm still fairly new to the topic. I understand finding the transforms very basic signals.

But the approach to finding the transform of this following one threw me off

How do I go about solving this?

Answer 1

Try starting with the defintion, treating the two domains as separate Z transforms that are being added together:

$$X_2(z) = \sum\limits_{n=-\infty}^{-1}\left(\dfrac{1}{2}\right)^{-n} z^{-n} + \sum\limits_{n=0}^{\infty}\left(\dfrac{1}{3}\right)^{n} z^{-n}$$

Some simple initial manipulations yield:

$$X_2(z) =-1 + \sum\limits_{n=0}^{\infty}\left(\dfrac{1}{2}\right)^{n} z^{n} + \sum\limits_{n=0}^{\infty}\left(\dfrac{1}{3}\right)^{n} z^{-n}$$