825 - Analysis

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.

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