Complex Power Series

Definition: Complex Power Series

A complex power series is an expression of the form

$$\sum _{n\in \mathcal{D}}{z}_n(x-c{)}^n,$$

where $(z_n{)}_{n\in \mathcal{D}}$ is a complex sequence and $c\in \mathbb{C}$.

INTUITION

By plugging in a concrete value $x^{\ast }\in \mathbb{C}$ for $x$, one obtains the complex series

$$\sum _{n\in \mathcal{D}}{z}_n(x^{\ast }-c{)}^n$$

Convergence

Definition: Convergence and Divergence of Power Series

Let $\sum _{n\in \mathcal{D}}{z}_n(x-c{)}^n$ be a complex power series and let $x^{\ast }\in \mathbb{C}$.

We say that $\sum _{n\in \mathcal{D}}{z}_n(x-c{)}^n$

• is convergent or converges for $x^{\ast }$ if the resultant complex series ${\sum }_{n\in \mathcal{D}}{z}_n(x^{\ast }-c{)}^n$ is convergent;
• is absolutely convergent or converges absolutely for $x^{\ast }$ if the resultant complex series ${\sum }_{n\in \mathcal{D}}{z}_n(x^{\ast }-c{)}^n$ is absolutely convergent;
• is divergent or diverges for $x^{\ast }$ if the resultant complex series ${\sum }_{n\in \mathcal{D}}{z}_n(x^{\ast }-c{)}^n$ is divergent;

Theorem: Radius of Convergence

There are only three possibilities each complex power series $\sum _{n\in \mathcal{D}}{z}_n(x-c{)}^n$:

• The power series converges only for $x=c$.
• The power series converges for all $x\in \mathbb{C}$.
• There exists some $r>0$ such that the power series converges if $|x-c|<r$ and diverges if $|x-c|>r$. If $|x-c|=r$, then the power series may or may not converge and it might do so only for some points but not for others.

PROOF

TODO

Definition: Radius of Convergence

If $r$ exists, then we call it the radius of convergence. Furthermore, if the power series converges only for $x=c$, we usually say that the radius of convergence is zero. If the power series converges for all $x\in \mathbb{C}$, then we say that the radius of convergence if infinite.

Algorithm: Determining the Radius of Convergence

We are given a complex power series $\sum _{n\in \mathcal{D}}{z}_n(x-c{)}^n$ and want to determine its radius of convergence.

1. Evaluate either one of the limits ${\lim }_{n\to \infty }\left|\frac{z_n}{z_{n+1}}\right|$ and ${\lim }_{n\to \infty }\frac{1}{\sqrt[n]{|z_n|}}$. Choose whichever one is easier to calculate.

2. If the limit is zero, then the power series converges only for $x=c$.

3. If the limit is $+\infty$, then the power series converges for every $x\in \mathbb{R}$.

4. If the limit is some nonzero real number, then it is the radius of convergence.