the disk of convergence
The disk of convergence is the actual region of the plane where a power series lives. If a power series sum a_n (z - z_0)^n has radius of convergence R, its disk of convergence is the open disk of all points z with |z - z_0| < R — a round patch centred at z_0. On this disk the series converges and defines a function; everywhere strictly outside it the series diverges. The disk, not some odd-shaped blob, is the natural home of every power series.
It matters that the disk is OPEN — it does not include its own boundary circle |z - z_0| = R. Inside the disk the convergence is excellent: not just convergent but absolutely convergent, and uniformly convergent on every smaller closed disk you draw inside. That uniform convergence on the interior is what makes the sum a genuinely holomorphic function and lets you differentiate and integrate term by term. The boundary circle is the murky frontier: there the series might converge at some points and diverge at others, and no single rule covers all cases. So we draw a firm line around the open disk and treat the rim separately.
Picture it geometrically for a Taylor series. The function you are expanding has some nearest singularity — a pole, a branch point, somewhere it blows up or stops being holomorphic. The disk of convergence is the largest open disk centred at z_0 that fits without touching that singularity; its boundary circle just grazes the nearest bad point. This is why the disk of convergence is sometimes called the largest disk of holomorphy around the centre: it is exactly the reach of the local power-series description before the function's global geography intrudes.
Expand Log(1 + z) = z - z^2/2 + z^3/3 - ... around z_0 = 0. The function has a branch point (a singularity) at z = -1, distance 1 from the centre, so the disk of convergence is the open unit disk |z| < 1. The series is useless for, say, z = 2; to evaluate Log there you re-expand around a centre closer to 2.
The disk reaches out exactly to the nearest singularity at z = -1; the boundary circle passes through that point.
Do not confuse the disk of convergence (where the series converges) with the domain of the function it represents. The function 1/(1 - z) is defined and holomorphic far outside the unit disk, but the particular series sum z^n only converges inside it.