Areas in developing laurent series

[nhrock3]

$$f(x)=\frac{-2}{z-1}$$+$$\frac{3}{z+2}$$
our distance is from -2 till 1
we develop around 1
so our distances are 3 and zeo
so our areas are
0<|z-1|<3
0<|z-1|
3>|z-1|
but i was told to develop around

0<|z-1|<1
there is no such area
?

 

[HallsofIvy]

0< |z- 1|< 1 is -1< z< 1 and also $z\ne 0$- two separate intervals.

 

[vela]

$$f(x)=\frac{-2}{z-1}$$+$$\frac{3}{z+2}$$
our distance is from -2 till 1
we develop around 1
so our distances are 3 and zeo
so our areas are
0<|z-1|<3
0<|z-1|
3>|z-1|
but i was told to develop around

0<|z-1|<1
there is no such area
?

The region 0<|z-1|<1 lies completely within 0<|z-1|<3, right? So if you find the Laurent series that converges in 0<|z-1|<3, it will obviously converge when 0<|z-1|<1.

 

[nhrock3]

thanks
:)