Power Series Basics

Complex Analysis · Axiom Academy

An infinite polynomial that converges inside a disk — and the radius that tells you exactly how big the disk is. 1. An Infinite Polynomial That Becomes a Function Add up the geometric power series one term at a time. Watch the running total — the partial sum — climb toward the smooth curve . As long as you stay where |z| < 1 , more terms means a better fit. Step past z = 1 and the partial sums fly apart: the series diverges . In the complex plane " |z| < 1 " isn't an interval — it's a disk . Every power series has a radius of convergence : a circle of radius R about the center z_0 that splits the plane in two. Drag a test point around and watch its terms |a_n(z-z_0)^n| : inside they shrink to zero and the series converges absolutely; outside they blow up and it diverges. |z - z_0| < R : the terms decay geometrically, so the series converges absolutely . |z - z_0| > R : the terms do not even tend to zero, so the series diverges . |z - z_0| = R : anything can happen — convergence must be checked point by point. z = z_0 always converges (every term after a_0 vanishes), so always. Absolute convergence inside the disk is what lets us differentiate and integrate a power series term by term — and the new series keeps the very same radius R . That is precisely why a power series defines an analytic (infinitely differentiable) function on its disk. 3. Measuring R With the Ratio Test

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