Difference between revisions of "Convergence of a sequence"
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Revision as of 17:12, 8 March 2015
Like with continuity there are three forms for convergence of a Sequence
Given a sequence [math](a_n)^n_{n=1}[/math] we may say that it converges to [ilmath]a[/ilmath] or [math]\lim_{n\rightarrow\infty}(a_n)=a[/math] if and only if the following definition holds:
Contents
First form
Introductory form [math]\forall\epsilon>0\exists N\in\mathbb{N}:n> N\implies|a_n-a|<\epsilon[/math]
Second form
Metric space form [math]\forall\epsilon>0\exists N\in\mathbb{N}:n> N\implies d(a_n-a)<\epsilon[/math]
Third form
Topological form [math]\forall N_a\exists N\in\mathbb{N}: n> N\implies a_n\in N_a[/math] where [math]N_a[/math] denotes a neighbourhood of [math]a[/math]
Cauchy Criterion
Convergence can be shown without knowing what exactly the sequence converges to, see the Cauchy criterion for convergence page
Note on norms
Recall from norm that we can simply define [math]d_{\|\cdot\|}(x,y)=\|x-y\|[/math], thus we can also have a slight variation of the metric form:
[math]\forall\epsilon>0\exists N\in\mathbb{N}:n> N\implies \|a_n-a\|<\epsilon[/math]
Is is worth noting because in Functional Analysis norms are considered and if we deal with a metric space we are inside a branch of topology
Interesting examples
[math]f_n(t)=t^n\rightarrow 0[/math] in [math]\|\cdot\|_{L^1}[/math]
Using the [math]\|\cdot\|_{L^1}[/math] norm stated here for convenience: [math]\|f\|_{L^p}=\left(\int^1_0|f(x)|^pdx\right)^\frac{1}{p}[/math] so [math]\|f\|_{L^1}=\int^1_0|f(x)|dx[/math]
We see that [math]\|f_n\|_{L^1}=\int^1_0x^ndx=\left[\frac{1}{n+1}x^{n+1}\right]^1_0=\frac{1}{n+1}[/math]
This clearly [math]\rightarrow 0[/math] - this is [math]0:[0,1]\rightarrow\mathbb{R}[/math] which of course has norm [ilmath]0[/ilmath], we think of this from the sequence [math](\|f_n-0\|_{L^1})^\infty_{n=1}\rightarrow 0\iff f_n\rightarrow 0[/math]