Difference between revisions of "825 - Analysis"
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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>. | '''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 | + | ===Completeness Axiom=== |
| + | An ordered field <math>\mathbb{F} </math> is ''complete'' if for any <math>E \sube \mathbb{F}, E\neq \emptyset </math>, if <math>E </math> is bounded above, <math>E </math> has at least upper bound. | ||
===Functional Limits=== | ===Functional Limits=== | ||
Latest revision as of 17:57, 14 December 2022
Contents
Sequences
- see also Sequences (Analysis)
Given \(X\neq \emptyset \) and \(k \in \Z \), a sequence from \( X\) is a function \(a:\{k,k+1,k+1,...\}\to X \).
Usual Notation: \(\{a_n\}_{n=k}^\infty \sube X \) or \( \{a_n\}\sube X\) denotes a sequence from \(X \) with \(a_n=a(n) \) for each \(n\in \{k,k+1,k+2,...\} \).
Completeness Axiom
An ordered field \(\mathbb{F} \) is complete if for any \(E \sube \mathbb{F}, E\neq \emptyset \), if \(E \) is bounded above, \(E \) has at least upper bound.
Functional Limits
Compactness
Sequential Compactness
Differentiation
Darboux
- see also Darboux's Theorem
If \(f:I\to \mathbb{R}\) is diff. on \(I\), then \(f'\) is Darboux on \(I\); i.e. \(f'\) has the initial value property.
Taylors Theorem
With \(n\in \mathbb{W}\) suppose \(f:I\to \mathbb{R}\) is \((n+1)\) times diff on \(I\). For each \(x,x_0 \in I\), there exists \(\xi\in I\backslash \{x\}\) between \(x\) and \(x_0\) such that \(f(x)=P_n(x)+\frac{1}{(n+1)!}f^{(n+1)}(\xi)(x-x_0)^{n+1}\)
- Remark\[\xi\] depends on \(x\) and need not be unique
