I am looking for the inverse $\mathcal Z$-transform of the following:
$$ \frac{1}{1-\frac 12 z^{-1}}+\frac{1}{1+\frac 13 z^{-1}} $$
When the region of convergence is $z > 1/3$. I have found the $\mathcal Z$-transform for when $z > 1/2$ as a right sided signal:
$$ \left(\frac{1}{2}\right)^n u[n] + \left(-\frac{1}{3}\right)^n u[n] $$
I can't seem to find a inverse $\mathcal Z$-transform when the ROC goes outwards from the inner pole. Does the inverse $\mathcal Z$-transform not exist for $z>1/3$?