For the Laurent series for $\displaystyle f(z)=(z-3)\sin{\frac{1}{z+2}}$ about $\displaystyle z=-2$.

The convergence region is the disc with center in z=-2 and $\displaystyle 0<radius<5$, since we have singularities in z=3 and z=-2.

Is this correct?

The textbook unswer is that the series converges for $\displaystyle \forall z\neq -2$