Find the region of convergence of the power series

where

.

According to my class notes, if I have a power series of the form

, the radius of convergence is equal to

. So I applied the formula and I got that

.

I wrote

. So I think that

depends on

. If

the series converges while if

the series diverges,

. So the radius of convergence would be 1, although it seems totally wrong to me since I think I should have had a condition on z, not only on a. For instance if

I fall in the case when I have

and the radius of convergence is indeed 1 but because of a condition on z.

So I'd like to know what I'm doing wrong.