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

# {{ page.title }}

{% definition Power Series %}
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.
{% enddefinition %}

{% 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 $\abs{z-a} < \abs{w}$.
{% endlemma %}

{% 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.
{% endtheorem %}

{% proof %}
TODO
{% endproof %}

{% theorem * 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

$$
\frac{1}{R} = \limsup_{n \to \infty} \norm{x_n}^{1/n}.
$$
{% endtheorem %}

{% proof %}
TODO
{% endproof %}