Difference between revisions of "Convergence of a sequence"
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Latest revision as of 13:30, 5 December 2015
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TODO: preserve "interesting" example
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]
