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

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 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 $z=0$.

Defines an analytic function

A power series defines an analytic function on its disk of convergence. You can differentiate and integrate it term by term inside that disk, and the radius of convergence stays the same.

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).

Related series

A Taylor series is a power series with a specific recipe for the coefficients. Laurent series generalize this by allowing negative powers, for functions with singularities.

The radius of convergence equals the distance from $z_0$ to the nearest singularity of the underlying analytic function. A geometric reading of an algebraic quantity.