Region of convergence

Summary

The region of convergence of a Z-transform is the set of complex numbers $z$ where its defining series converges absolutely. It is an annulus centred at $0$, possibly empty, possibly with some boundary points. It plays the part that the disc given by the radius of convergence plays for a power series. A Z-transform is only known together with its region of convergence: two different sequences can have the same formula for $X(z)$ on two different regions. The position of the region with respect to the unit circle tells whether a system is causal and stable.