825 - Analysis - Revision history

Kfagerstrom: /* Completeness Axioms */

2022-12-14T17:57:39Z

Completeness Axioms

Kfagerstrom: /* Sequences */

2022-12-07T20:02:35Z

Sequences

Kfagerstrom: Created page with " ===Sequences=== ===Completeness Axioms=== ===Functional Limits=== ===Compactness=== ====Sequential Compactness==== ===Differentiation=== ====Darboux==== :''see also'' [..."

2022-12-07T19:51:49Z

Created page with " ===Sequences=== ===Completeness Axioms=== ===Functional Limits=== ===Compactness=== ====Sequential Compactness==== ===Differentiation=== ====Darboux==== :''see also'' [..."

New page

===Sequences===

===Completeness Axioms===

===Functional Limits===

===Compactness===
====Sequential Compactness====

===Differentiation===

====Darboux====
:''see also'' [[Darboux's Theorem]]
If <math>f:I\to \mathbb{R}</math> is diff. on <math>I</math>, then <math>f'</math> is ''Darboux'' on <math>I</math>; i.e. <math>f'</math> has the initial value property.

====Taylors Theorem====

With <math>n\in \mathbb{W}</math> suppose <math>f:I\to \mathbb{R}</math> is <math>(n+1)</math> times diff on <math>I</math>. For each <math>x,x_0 \in I</math>, there exists <math>\xi\in I\backslash \{x\}</math> between <math>x</math> and <math>x_0</math> such that <math>f(x)=P_n(x)+\frac{1}{(n+1)!}f^{(n+1)}(\xi)(x-x_0)^{n+1}</math>

:Remark: <math>\xi</math> depends on <math>x</math> and need not be unique

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 '''Usual Notation:''' <math>\{a_n\}_{n=k}^\infty \sube X </math> or <math> \{a_n\}\sube X</math> denotes a sequence from <math>X </math> with <math>a_n=a(n) </math> for each <math>n\in \{k,k+1,k+2,...\} </math>.
-===Completeness Axioms===
+===Completeness Axiom===
 ===Functional Limits===

← Older revision		Revision as of 20:02, 7 December 2022
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	===Sequences===		===Sequences===
		+	:''see also'' [[Sequences (Analysis)]]
		+
		+	Given <math>X\neq \emptyset </math> and <math>k \in \Z </math>, a ''sequence'' from <math> X</math> is a function <math>a:\{k,k+1,k+1,...\}\to X </math>.
		+
		+	'''Usual Notation:''' <math>\{a_n\}_{n=k}^\infty \sube X </math> or <math> \{a_n\}\sube X</math> denotes a sequence from <math>X </math> with <math>a_n=a(n) </math> for each <math>n\in \{k,k+1,k+2,...\} </math>.

	===Completeness Axioms===		===Completeness Axioms===