File:Ratio test proof.svg
Original file (SVG file, nominally 201 × 134 pixels, file size: 56 KB)
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Summary
[edit]DescriptionRatio test proof.svg |
English: A plot showing how the ratio test test is proven in the convergent case. Given a sequence like the blue one, for which the ratio of adjacent terms converges to L < 1, we identify a ratio r = (L+1)/2 and show that for large enough n the sequence is dominated by the simple geometric sequence rk. In this case the ratio of adjacent terms of the blue sequence converges to L=1/2, so we choose r=3/4, and rk dominates for all n ≥ 2. Source used to generate this chart is shown below. |
Date | |
Source |
Own work This W3C-unspecified diagram was created with Mathematica. |
Author | User:Dcoetzee |
Licensing
[edit]This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication. | |
The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.
http://creativecommons.org/publicdomain/zero/1.0/deed.enCC0Creative Commons Zero, Public Domain Dedicationfalsefalse |
Source
[edit]All source released under CC0 waiver.
Mathematica source to generate graph (which was then saved as SVG from Mathematica):
Show[][][(3/4)^n, {n, 1, 15}], PlotStyle -> {Hue[0], PointSize[Large]}], ListPlot[][(n + 1)/n * (1/2)^n, {n, 1, 15}], PlotStyle -> PointSize[Large], PlotRange -> All, AxesStyle -> FontSize -> 14]
LaTeX source for labels:
$$ \left(\frac{3}{4}\right)^n $$ $$ \frac{n+1}{n}\left(\frac{1}{2}\right)^n $$
These were converted to SVG with [1] and then the graph was embedded into the resulting document in Inkscape. Axis fonts were also converted to Liberation Serif and the legend was added in Inkscape.
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 17:31, 24 March 2013 | 201 × 134 (56 KB) | Dcoetzee (talk | contribs) | == {{int:filedesc}} == {{Information |Description ={{en|1=A plot showing how the en:ratio test test is proven in the convergent case. Given a sequence like the blue one, for which the ratio of adjacent terms <math>|a_{n+1}/a_n|</math> converge... |
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Width | 201.4668 |
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Height | 133.53326 |