Laurent Series - when do singularities on the boundary of an annulus require a Laurent series instead of Taylor?

Question

I need to find the Laurent Expansion of $F(z) = \dfrac{1}{(z-1)^2(z+2)}$ in the regions $A_1 = D(0,1)$ and $A_2 = \{z: 1 < |z| < 2 \}$.

After doing partial fractions on $F(z)$, how do I know whether I need to work out a regular Taylor series or one of those fancy Laurent series, since the singularities aren't actually in the annulus but on the (open) boundary?

Thanks

Answer 1

If the summand is regular in the point (no $1/0$), has a Taylor series.