Region of Convergence z-transform (ROC) Given the z-transform of a sequence: > $$ X(z) = \frac{1-\frac{1}{2}z^{-1}}{1+\frac{3}{4}z^{-1}+\frac{1}{8}z^{-2}} , |z| > \frac{1}{2} $$ after partial fraction decomposition I get, > $$ X(z) = \frac{4}{1+\frac{1}{2}z^{-1}} + \frac{-3}{1+\frac{1}{4}z^{-1}} $$ Now I was thinking about the ROC (region of convergence). I know from the given that $|z| > \frac{1}{2}$, but now I have a pole at $\frac{1}{4}$. What will my ROC be / look like?

Your $X(z)$ has two poles, at $z=1/2$ and $z=1/4$, hence there are, a priori, three possible ROC: inside the inner circle ($|z|<1/4$), the annulus between the two circles ($1/4<|z|<1/2$) and the region outside the exterior circle ($|z|>1/2$). But you are already told that you must consider the later ROC. So all it's right, you just go ahead.

(You had reasons to worry -to suspect that something was wrong- if you had obtained a pole at, say $z=3/4$)