find the laurent series of $\displaystyle f(x)=\frac{-2}{z-1}$+$\displaystyle \frac{3}{z+2}$

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

$\displaystyle f(x)=\frac{1}{1-z}$

then

the radius is 1 and

because 1-1=0

so

it is analitical on

|z|<1

so if i apply the same logic

$\displaystyle f(x)=\frac{-2}{z-1}$

1 still makes denominator 0

and

it is analitical on

|z|<1

but the correct answer is

it is analitical on

|z|>1

for

$\displaystyle f(x)=\frac{3}{z+2}$

-2 makes denominator 0

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

where is my mistake?