File:Ratio test proof.svg

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Original file(SVG file, nominally 201 × 134 pixels, file size: 56 KB)

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Summary

Description
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

I, the copyright holder of this work, hereby publish it under the following license:
Creative Commons CC-Zero 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.

Source

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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24 March 2013

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Date/TimeThumbnailDimensionsUserComment
current11:31, 24 March 2013Thumbnail for version as of 11:31, 24 March 2013201 × 134 (56 KB)Dcoetzee== {{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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