Is the Hilbert series of graded module interpreted as a rational function meaningful everywhere?

Question

Say we have a graded module $M$ over a field $k$. In my mind I have $M=k[x,y,z]/(y^2-xz)$ graded by rank. The Hilbert series of this is $\sum_{n=0}^{\infty} (2n+1)t^n$. You can represent this as a rational function $(1+t)/(1-t)^2$. Now, I computed this as a Taylor series so my interval of convergence is $(-1,1)$. Can I use facts about the rational function outside of that interval to prove things about $M$?