From 28407333ffceca9b99fae721c30e8ae146a863da Mon Sep 17 00:00:00 2001
From: Justin Gassner <justin.gassner@mailbox.org>
Date: Wed, 14 Feb 2024 07:24:38 +0100
Subject: Update

---
 .../one-complex-variable/power-series.md           | 76 +++++++++++-----------
 1 file changed, 37 insertions(+), 39 deletions(-)

(limited to 'pages/complex-analysis/one-complex-variable/power-series.md')

diff --git a/pages/complex-analysis/one-complex-variable/power-series.md b/pages/complex-analysis/one-complex-variable/power-series.md
index 0147f31..31793ab 100644
--- a/pages/complex-analysis/one-complex-variable/power-series.md
+++ b/pages/complex-analysis/one-complex-variable/power-series.md
@@ -3,59 +3,57 @@ 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 $\abs{z-a} < \abs{w}$.
+{% 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.
+{% 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-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
-> 
-> $$
-> \frac{1}{R} = \limsup_{n \to \infty} \norm{x_n}^{1/n}.
-> $$
+{% 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
-- 
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