So, I will say it again..
Writing Z=.. and then assuming it is a well defined quantity is an assumption you cannot make.
The sum is nonconvergent. However you can very simply analytically continue the function that the sum represents beyond the radius of convergence but that is a continuation. The sum itself is not defined, it is not convergent. So if its nonconvergent don't write Z=..
It doesn't equal anything...