Power Series and the Radius of Convergence
From the geometric series prototype to the full convergence toolkit
1. What Is a Power Series?
A power series is an infinite series whose terms are coefficients multiplied by powers of a variable. In its simplest (origin-centered) form:
More generally, a power series centered at a point $a$ (the shifted form) is:
Here each $a_n$ is a fixed real number (the coefficient) that may depend on $n$ but not on $x$. The point $a$ is called the center of the series.
🔑 The Key Idea: A Machine That Turns Points into Series
Think of a power series as a machine. You feed it a number $x$, and it produces a numerical series — a sum of constants — that either converges or diverges. The central question is:
The Geometric Power Series — Our Prototype
The simplest and most important power series is the geometric series:
From our earlier study of geometric series, we know this converges precisely when $|x| < 1$ and diverges when $|x| \geq 1$. The convergence set is the open interval $(-1, 1)$.
This tells us the radius of convergence is $R = 1$. The series converges for all $x$ within distance $1$ of the center (the origin), and diverges for all $x$ farther away. This is the template for everything that follows.
2. The Structure Theorem and the Trichotomy
Why the Convergence Set Is Always an Interval
A remarkable theorem tells us that the convergence set of any power series is not some random collection of points — it is always an interval centered at the center of the series.