---
title: Power Series
parent: One Complex Variable
grand_parent: Complex Analysis
nav_order: 1
# cspell:words
---

# {{ page.title }}

{: .definition-title }
> Definition ({{ page.title }})
>
> Let $X$ be a complex Banach space.
> A *power series* (with values in $X$) is an infinite series of the form
>
> 
> $$
> \sum_{n=0}^{\infty} x_n (z - a)^n = x_0 + x_1 (z-a) + x_2 (z-a)^2 + \cdots,
> $$
>
> where $x_n \in X$ is the *$n$th coefficient*,
> $z$ is a complex variable and
> $a$ is the *center* of the series.

{: .lemma }
> Suppose the Banach space valued power series $\sum_{n=0}^{\infty} x_n (z - a)^n$ converges for $z = a + w$.
> Then it converges absolutely for all $z$ with $\abs{z-a} < \abs{w}$.

{% proof %}
TODO
{% endproof %}

{: .theorem }
> Suppose $\sum_{n=0}^{\infty} x_n (z - a)^n$ is a Banach space valued power series.
> Then either
>
> - the series converges only for $z=a$ (formally $R=0$), or
> - there exists a number $0<R<\infty$ such that
>   the series converges absolutely whenever $\abs{z-a} < R$ 
>   and diverges whenever $\abs{z-a} > R$, or
> - the series converges absolutely for all $z \in \CC$ (formally $R=\infty$).
>
> The number $R \in [0,\infty]$ is called the *radius of convergence* of the power series.

{% proof %}
TODO
{% endproof %}

{: .theorem-title }
> Cauchy–Hadamard Formula
>
> Let $\sum_{n=0}^{\infty} x_n (z - a)^n$ be a Banach space valued power series
> with radius of convergence $R$. Then
> 
> $$
> \frac{1}{R} = \limsup_{n \to \infty} \norm{x_n}^{1/n}.
> $$

{% proof %}
TODO
{% endproof %}