Radius of Convergence

Complex Analysis · Axiom Academy

A power series doesn't converge on an interval — it converges on a disk . One number, the radius R, draws the line between order and chaos. 1. Convergence Lives on a Disk Center a power series at z_0 and ask: for which z does add up to a finite number? The answer has a startlingly clean shape. There is a single radius that splits the whole plane into converge inside and diverge outside. Inside the disk — converges absolutely 2. The Coefficients Set the Radius You never have to test points one at a time — the coefficients tell you R directly. Feed in the ratios ; whatever number they settle toward is the radius. The Cauchy–Hadamard root test below always works, even when the ratios misbehave. , so R = 0 — the series converges only at the center. The ratio is exactly 2 at every n , so R = 2 — convergence for |z| < 2 . , so — converges on the whole plane (an entire function, here e^z ). When ratios bounce around, still pins down R exactly. For the ratio limit is , so and the series converges precisely for . 3. The Radius Reaches the Nearest Singularity Here is why power series are a complex story. The disk grows outward from z_0 until it collides with the nearest point where the function blows up . That collision sets R . A singularity sitting off the real axis still caps the radius — which is invisible from real calculus alone. The clean case — : the only singularity is the pole at z = 1 . It lies a distance 1 from the center, and indeed R = 1 .

This is the written version of the interactive lesson above. See the full Complex Analysis course.