Series: ratio test

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Ben Brawn

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7 years, 7 months ago

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Ben Brawn 7 years, 7 months ago

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Ben Brawn 7 years, 7 months ago

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Ben Brawn 7 years, 7 months ago

Created this as a copy of Series: divergence test.

Name	Status	Author	Last Modified
Series: divergence test	Ready to use	Ben Brawn	03/10/2017 07:52
Series: ratio test	Ready to use	Ben Brawn	03/10/2017 07:50
Series: comparison test	Ready to use	Ben Brawn	03/10/2017 07:49
Monotonic sequence theorem	Ready to use	Ben Brawn	03/10/2017 07:48

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Question statement

This question is about the ratio test for series.

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			Name	Type	Generated Value

seed	integer	0
percent	integer	100
limit	rational	17/10
a	integer	3
d	integer	6
coeff	list	[ -10, 0, 3, -6, 0 ]
b	number	-10
c	integer	0
f	number	3
g	number	-6
h	integer	0
alt	integer	1
cseed	number	0.5
cexpression	string	$\displaystyle\simplify{(-1)^k
ratio_con_to_zero	list	List of 3 items
ratio_con_to_less_than_one	list	List of 10 items
ratio_con_to_one	list	List of 10 items
ratio_con_to_big	list	List of 10 items
ratio_div_to_inf	list	List of 2 items
start	integer	4

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A series is such that as we get further and further along the sequence, each term has an absolute value that approaches $\var{percent}\%$ of the absolute value of the previous term. What does the ratio test tell us about this series?

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This series converges.

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This series diverges.

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The ratio test says, given a series $\sum_{k=a}^\infty t_k$

if $\lim_{k\rightarrow\infty}\left|\frac{t_{k+1}}{t_k}\right|<1$ , then the series converges
if $\lim_{k\rightarrow\infty}\left|\frac{t_{k+1}}{t_k}\right|>1$ , then the series diverges
if $\left|\frac{t_{k+1}}{t_k}\right|\rightarrow\infty$ as $k\rightarrow \infty$ , then the series diverges

Notice, if the limit doesn't exist for some other reason or is equal to $1$ the test doesn't tell us anything, some might say the test is 'inconclusive' or 'fails'.

Since $\lim_{k\rightarrow\infty}\left|\frac{t_{k+1}}{t_k}\right|=\frac{\var{percent}}{100}<1$ the ratio test tells us that the series converges.

Since $\lim_{k\rightarrow\infty}\left|\frac{t_{k+1}}{t_k}\right|=\frac{\var{percent}}{100}>1$ the ratio test tells us that the series diverges.

Since $\lim_{k\rightarrow\infty}\left|\frac{t_{k+1}}{t_k}\right|=1$ the ratio test doesn't tell us anything.

Since $\lim_{k\rightarrow\infty}\left|\frac{t_{k+1}}{t_k}\right|=\var[fractionNumbers]{limit}<1$ the ratio test tells us that the series converges.

Since $\lim_{k\rightarrow\infty}\left|\frac{t_{k+1}}{t_k}\right|=\var[fractionNumbers]{limit}>1$ the ratio test tells us that the series diverges.

Since $\lim_{k\rightarrow\infty}\left|\frac{t_{k+1}}{t_k}\right|=1$ the ratio test doesn't tell us anything.

To use the ratio test we need to determine the value that the absolute value of the ratio of consecutive terms converge to.

Given $t_k=$ {cexpression} we use algebra and limit laws to determine

$\lim_{k\rightarrow\infty}\frac{t_{k+1}}{t_k}=0$

$\lim_{k\rightarrow\infty}\frac{t_{k+1}}{t_k}=\simplify[fractionNumbers]{{1/a}}$

$\lim_{k\rightarrow\infty}\frac{t_{k+1}}{t_k}=1$

$\lim_{k\rightarrow\infty}\frac{t_{k+1}}{t_k}=\var{a}$

and therefore by the ratio test the series converges.

and therefore by the ratio test the series diverges.

and therefore the ratio test doesn't tell us anything.

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This question is used in the following exams:

Series by Ben Brawn in Sequences and Series.
analysis by Daniel Atcheson in Daniel's workspace.
Series Homework 1 by Ugur Efem in Ugur's workspace.

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Ben Brawn 7 years, 7 months ago

Ben Brawn 7 years, 7 months ago

Ben Brawn 7 years, 7 months ago

Error in variable testing condition

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Series: ratio test

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

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Ben Brawn 7 years, 7 months ago

Ben Brawn 7 years, 7 months ago

Ben Brawn 7 years, 7 months ago

Error in variable testing condition

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Generated value: integer

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Generated value: `integer`