Complex Power Series — Mathematics Knowledge System

Overview

A complex power series is an infinite series of the form Σ aₙ(z − z₀)ⁿ, where the coefficients aₙ and center z₀ are complex numbers and z ranges over the complex plane. Just as with real power series, every complex power series has a radius of convergence R ≥ 0 (possibly infinite): the series converges absolutely for every z with |z − z₀| < R and diverges for every z with |z − z₀| > R, while the boundary circle |z − z₀| = R is a genuine circle of ambiguity where convergence must be checked point by point. What makes the complex setting so much richer than the real line is geometric: the region of guaranteed convergence is an actual disk in the plane, not just an interval, and a function defined by a convergent power series is automatically holomorphic inside that disk, with a derivative obtained by differentiating the series term by term. This is the concrete computational bridge between formal Taylor series and the abstract notion of an analytic function.

Intuition

On the real line, a power series converges on an interval because the terms aₙxⁿ eventually shrink once |x| is small enough relative to how fast |aₙ| grows. In the complex plane the exact same growth-rate argument applies, but distance now has two dimensions to move in, so the 'safe zone' is a disk instead of an interval — the radius of convergence R is genuinely the radius of a circle centered at z₀. Nothing about the argument privileges the real axis: it depends only on |z − z₀|, the plain distance from the center. This is also why a power series can converge everywhere inside a disk yet be forced to fail somewhere on the boundary circle — a singularity of the sum function anywhere on that circle caps how far the disk of guaranteed convergence can extend, even in directions where nothing obviously goes wrong.

Formal Definition

Definition

A complex power series centered at z₀ ∈ ℂ with coefficients aₙ ∈ ℂ is the formal sum:

\sum_{n=0}^{\infty} a_n (z - z_0)^n

Complex power series

\frac{1}{R} = \limsup_{n\to\infty} |a_n|^{1/n}

Cauchy–Hadamard formula for the radius of convergence

R = \lim_{n\to\infty} \left|\frac{a_n}{a_{n+1}}\right| \quad \text{(when the limit exists)}

Ratio-test shortcut for R

Worked Examples

Here aₙ = 1 for all n, so |aₙ|^{1/n} = 1 for all n.
$|a_n|^{1/n} = 1^{1/n} = 1$
Apply the Cauchy–Hadamard formula.
$\frac{1}{R} = \limsup_{n\to\infty} 1 = 1 \;\Rightarrow\; R = 1$

Answer: R = 1: the series converges for |z| < 1 (to 1/(1−z)) and diverges for |z| > 1.

Practice Problems

Difficulty 5/10

Find the radius of convergence of Σ_{n=1}^∞ zⁿ/n.

Difficulty 6/10

Find the radius of convergence of Σ_{n=0}^∞ n! zⁿ.

Difficulty 7/10

The function f(z) = 1/(1+z²) has singularities at z = i and z = −i. Without computing any coefficients, explain why its Taylor series centered at z₀ = 0 must have radius of convergence exactly 1.

Quiz

Summary

A complex power series Σ aₙ(z−z₀)ⁿ converges absolutely inside a disk |z−z₀| < R and diverges outside it, with R given by 1/R = limsup |aₙ|^{1/n} (Cauchy–Hadamard).
Convergence on the boundary circle |z−z₀| = R is not guaranteed and must be checked separately at each point.
A function defined by a convergent power series is holomorphic inside the disk of convergence and can be differentiated term by term.
For a Taylor series of an analytic function, R equals the distance from the center to the nearest singularity of the function.

References

WebsiteWikipedia — Power series
WebsiteWikipedia — Radius of convergence