Understanding Z Transform Convergence and Power Series for Stable Systems

[LM741]

hi guys.
Just want to clarify something with the z transform and convergence.

Do you guys aggree with this.

If an expression does not converge you CAN STILL find the z transform (using power series) for the system HOWEVER this will not be the closed form z transform.

just wondering - because in any z transform table you are given a region of convergence - but this is for closed form z transforms.
I suppose all systems are stable if they converge.

thanks

 

[LM741]

sorry - i think my last statement may be a bit to generalized:"I suppose all systems are stable if they converge."

you can get a system that converges but is not necessarily stable. E.g.
z/(z-a) where region of convergence is defined by, mod |z| > |a|

but a can be bigger then unity (which wil make the system unstable) but the system will still converge as long as z > a. Do you guys aggree?
makes sense beacuse i just read somewhere that for a system to be stable , the region of convergence must include the unit circle.
thanks