From 777f9d3fd8caf56e6bc6999a4b05379307d0733f Mon Sep 17 00:00:00 2001
From: Justin Gassner <justin.gassner@mailbox.org>
Date: Tue, 12 Sep 2023 07:36:33 +0200
Subject: Initial commit

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+---
+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 %}
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