convergence in an annulus

A Taylor series converges inside a disk — a filled-in circle. A Laurent series, because it carries both positive and negative powers, converges instead inside a ring: the region between two circles, like a washer or a donut seen from above. This shape is called an annulus, and understanding why is the whole story of where a Laurent series lives.

Split the series in two. The analytic part, sum over n greater than or equal to 0 of a_n (z - z_0)^n, is an ordinary power series; it converges inside some disk |z - z_0| < R (an outer circle of radius R). The principal part, sum over the negative powers, is a power series in the variable 1/(z - z_0); it converges when 1/(z - z_0) is small, that is when |z - z_0| > r for some inner radius r (outside an inner circle). For both halves to converge at once you need r < |z - z_0| < R — the annulus. There the convergence is even uniform on any closed sub-ring, so you may integrate and differentiate the series term by term.

The radii r and R are set by the nearest singularities: R reaches outward to the closest trouble spot away from z_0, while r shrinks inward to enclose just the singularity at z_0 itself (taking r = 0 gives a punctured disk 0 < |z - z_0| < R, the case you use to classify an isolated singularity). A common mistake is to expect one Laurent series for a function; really there is one per annulus, and a function with two singularities gives genuinely different expansions in the inner ring and the outer ring.