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.