radius of convergence
Picture a power series as a signal broadcast from its center. Close to the center the signal is strong and the series converges; far away it breaks down and diverges. The radius of convergence measures how far that good behavior reaches — the distance from the center out to where convergence finally fails.
Precisely, for a power series sum c_n (x - a)^n there is a number R in [0, infinity], the radius of convergence, with the property: the series converges absolutely whenever |x - a| < R and diverges whenever |x - a| > R. The cases R = 0 (converges only at the center) and R = infinity (converges everywhere) are allowed. The set |x - a| < R is the open interval of convergence on the real line, or the open disk of convergence in the complex plane.
The radius is determined entirely by the coefficients, via the Cauchy–Hadamard formula. What the radius does NOT tell you is what happens exactly on the boundary |x - a| = R: there the series may converge at some points, diverge at others, and no single rule covers every case. Those endpoints must be examined one at a time.
For sum x^n / n the ratio test gives |x| < 1, so R = 1. At x = 1 it is the harmonic series (diverges); at x = -1 it is the alternating harmonic series (converges). Same radius, opposite endpoint behavior.
Radius 1, but the two endpoints behave differently.
Why a radius and not some odd shape? Because absolute convergence at one point forces convergence at every closer point (Abel's lemma): if sum c_n z^n converges, the terms are bounded, and a geometric comparison handles all smaller |x|. That symmetry is exactly what produces a circular region.