Power series

A power series centered at $z_0\in \mathbb{C}$ is an infinite sum

${\sum }_{n=0}^{\infty }{a}_n(z-z_0{)}^n,$

with complex coefficients $a_n$. Each $z$ either makes the series converge or diverge.

Radius of convergence

Theorem. For each power series there is an $R\in [0,\infty ]$ — the radius of convergence — such that the series

• converges absolutely for $|z-z_0|<R$,
• diverges for $|z-z_0|>R$.

The boundary $|z-z_0|=R$ is case-by-case (can converge, diverge, or partly both).

The set $\{z:|z-z_0|<R\}$ is called the disk of convergence.

Computing $R$

Via the ratio test:

$\frac{1}{R}={\lim }_{n\to \infty }\left|\frac{a_{n+1}}{a_n}\right|,$

when the limit exists. If $|a_{n+1}/a_n|\to 0$, then $R=\infty$ (converges everywhere). If $|a_{n+1}/a_n|\to \infty$, then $R=0$ (converges only at $z_0$).

Sometimes the root test is easier: $1/R=\mathrm{limsup}|a_n{|}^{1/n}$.

Examples

Geometric series. $\sum z^n$. $a_n=1$, ratio $=1$, so $R=1$. Converges on $|z|<1$ to $1/(1-z)$.

Exponential. $\sum z^n/n!$. Ratio $|a_{n+1}/a_n|=1/(n+1)\to 0$. $R=\infty$. Converges everywhere to $e^z$.

Factorial. $\sum n!z^n$. Ratio $(n+1)!/n!=n+1\to \infty$. $R=0$. Converges only at $z=0$.

Key fact

A power series defines an analytic function on its disk of convergence. The function can be differentiated and integrated term by term within that disk, with the same radius of convergence preserved.

So power series are analytic functions, and analytic functions can be represented by power series (their Taylor series). The two viewpoints are equivalent on the disk of analyticity / convergence.

Convergence on the boundary

Whether a power series converges on the circle $|z-z_0|=R$ depends on the specific series:

• $\sum z^n$: diverges everywhere on $|z|=1$ (each term has $|z{|}^n=1$, doesn’t go to zero).
• $\sum z^n/n$: converges everywhere on $|z|=1$ except at $z=1$.
• $\sum z^n/n^2$: converges everywhere on $|z|=1$.

The boundary behavior is delicate and often physically important (Gibbs phenomenon, edge effects).

In context

Taylor series of analytic functions are power series with a specific recipe for the coefficients. Laurent series are a generalization that allows negative powers, used for functions with singularities. Together these power-series tools provide the most practical way to represent and compute with analytic functions.

The radius of convergence equals the distance from $z_0$ to the nearest singularity of the underlying analytic function — a beautiful geometric interpretation of an algebraic quantity. See Taylor series.