find the laurent series of +

for

1<|z|<2

i was by my teacher that the radius of convergence

is what smaller then the number which makes the denominator 0.

if

then

the radius is 1 and

because 1-1=0

so

it is analitical on

|z|<1

so if i apply the same logic

1 still makes denominator 0

and

it is analitical on

|z|<1

but the correct answer is

it is analitical on

|z|>1

for

-2 makes denominator 0

so |z|<-2 (but its illogical because |z| is a positive numbe)

where is my mistake?