Series: geometric series

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Xiaodan Leng 5 years, 10 months ago

Created this as a copy of Series: geometric series.

Name	Status	Author	Last Modified
Series: p series	Ready to use	Ben Brawn	03/10/2017 07:51
Series: geometric series	Ready to use	Ben Brawn	03/10/2017 07:51
Series: alternating series	Ready to use	Ben Brawn	03/10/2017 07:50
Series: geometric series	draft	Xiaodan Leng	11/07/2019 01:29

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Question statement

You are given the series

$\sum_{k=\var{start}}^\infty\var{a}\left(\var[fractionNumbers]{r}\right)^k.$

Give any introductory information the student needs.

			Name	Type	Generated Value

r	rational	-1/4
a	integer	-19
start	integer	7

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xxxxxxxxxx
 
random([1/3,1/4,1/6,-1/3,-1/4,-1/6,2/3,-2/3,3/4,-3/4,-2,-1.9,-1.8,-1.7,-1.6,-1.5,-1.4,-1.3,-1.2,-1.1,-1,-0.9,-0.8,-0.7,-0.6,-0.5,-0.4,-0.3,-0.2,-0.1,-2,1.9,1.8,1.7,1.6,1.5,1.4,1.3,1.2,1.1,0.9,0.8,0.7,0.6,0.5,0.4,0.3,0.2,0.1])

JME function reference

Description

Describe what this variable represents, and list any assumptions made about its value.

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Generated value: `rational`

-1/4

This variable doesn't seem to be used anywhere.

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Advice

A geometric series $\sum_{n=0}^\infty ar^n$ is convegent if $|r|<1$ with a sum of $\frac{a}{1-r}$ , and is divergent if $|r|\ge1$ .

Our series is $\sum_{k=\var{start}}^\infty\var{a}\left(\var[fractionNumbers]{r}\right)^k$ , which looks similar with an $r$ value of $\var[fractionNumbers]{r}$ . In fact, if we are worried that our series starts at $k=3$ instead of at $n=0$ notice:

$\displaystyle\sum_{k=\var{start}}^\infty \var{a}\left(\var[fractionNumbers]{r}\right)^k=\sum_{k=\var{start}}^\infty \var{a}\left(\var[fractionNumbers]{r}\right)^{\var{start}}\left(\var[fractionNumbers]{r}\right)^{k-\var{start}}$

we now make the substitution $n=k-\var{start}$ (which means when $k=\var{start}$ , $n=0$ ) and we have

$\displaystyle\sum_{n=0}^\infty \var{a}\left(\var[fractionNumbers]{r}\right)^{\var{start}}\left(\var[fractionNumbers]{r}\right)^{n}$

and so even though our series seemed to start later on, it is still a geometric series with $a=\var{a}\left(\var[fractionNumbers]{r}\right)^{\var{start}}$ and $r=\var[fractionNumbers]{r}$ . Also, note that adding or subtracting a finite number of terms to a series will not change whether it converges or diverges, nor will multiplying or dividing the series by a non-zero constant.

So we have a geometric series with common ratio $r=\var[fractionNumbers]{r}$ .

Since this common ratio is negative, each time we multiply by it we alternate the sign of the term, that is, this series is also an alternating series. If we wanted to, we could pull the common factor of $-1$ out and write our series as a positive number multiplied by $(-1)^n$ :

$\displaystyle\sum_{n=0}^\infty (-1)^n\var{abs(a)}\left(\var[fractionNumbers]{abs(r)}\right)^{\var{start}}\left(\var[fractionNumbers]{abs(r)}\right)^{n}$

The alternating series $\sum_{n=0}^\infty (-1)^n b_n$ where $b_n>0$ , converges if $b_{n+1}\le b_n$ for all $n$ and $\lim_{n\rightarrow\infty}b_n=0$ .

$\displaystyle\sum_{n=0}^\infty (-1)^{n+1}\var{abs(a)}\left(\var[fractionNumbers]{abs(r)}\right)^{\var{start}}\left(\var[fractionNumbers]{abs(r)}\right)^{n}$

The alternating series $\sum_{n=0}^\infty (-1)^{n+1} b_n$ where $b_n>0$ , converges if $b_{n+1}\le b_n$ for all $n$ and $\lim_{n\rightarrow\infty}b_n=0$ .

Therefore, we also know our series converges based on this alternating series test.

Therefore, we also know our series diverges based on this alternating series test.

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Taxonomy: mathcentre

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Xiaodan Leng 5 years, 10 months ago

Error in variable testing condition

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Generated value: `rational`

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Series: geometric series

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Xiaodan Leng 5 years, 10 months ago

Error in variable testing condition

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Warning

Generated value: rational

Parts

Gap-fill

Generated value: `rational`