Series: p series

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7 years, 7 months ago

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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

$\sum_{k=\var{start}}^\infty \simplify[fractionNumbers]{{num}/({den}*k^{p})}.$

Give any introductory information the student needs.

			Name	Type	Generated Value

p	rational	1/2
num	integer	13
den	integer	5
start	number	4

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(1,2,3,4,5,6,7,8,1/2,1/3,1/4,1/5,1/6,1/7,1/8)

JME function reference

Description

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

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Generated value: `rational`

1/2

This variable doesn't seem to be used anywhere.

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2. Gap Gap 1. Choose one from a list
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Advice

The $p$ -series $\sum_{n=1}^\infty \frac{1}{n^p}$ is convergent if $p>1$ and divergent if $p\le 1$ .

Our series is $\sum_{k=\var{start}}^\infty \simplify[fractionNumbers]{{num}/({den}*k^{p})}$ , which looks similar with a $p$ -value of $\var[fractionNumbers]{p}$ . In fact:

$\begin{align}\sum_{k=\var{start}}^\infty \simplify[fractionNumbers]{{num}/({den}*k^{p})}&=\simplify[fractionNumbers]{{num}/{den}}\sum_{k=\var{start}}^\infty \simplify[fractionNumbers]{1/(k^{p})}\\&=\simplify[fractionNumbers]{{num}/{den}}\left(-1-\frac{1}{2^\var[fractionNumbers]{p}}+\sum_{k=1}^\infty \simplify[fractionNumbers]{1/k^{p}}\right)\end{align}$

$\begin{align}\sum_{k=\var{start}}^\infty \simplify[fractionNumbers]{{num}/({den}*k^{p})}&=\simplify[fractionNumbers]{{num}/{den}}\sum_{k=\var{start}}^\infty \simplify[fractionNumbers]{1/(k^{p})}\\&=\simplify[fractionNumbers]{{num}/{den}}\left(-1-\frac{1}{2^\var[fractionNumbers]{p}}-\frac{1}{3^\var[fractionNumbers]{p}}+\sum_{k=1}^\infty \simplify[fractionNumbers]{1/k^{p}}\right)\end{align}$

$\begin{align}\sum_{k=\var{start}}^\infty \simplify[fractionNumbers]{{num}/({den}*k^{p})}&=\simplify[fractionNumbers]{{num}/{den}}\sum_{k=\var{start}}^\infty \simplify[fractionNumbers]{1/(k^{p})}\\&=\simplify[fractionNumbers]{{num}/{den}}\left(-1-\frac{1}{2^\var[fractionNumbers]{p}}-\frac{1}{3^\var[fractionNumbers]{p}}-\frac{1}{4^\var[fractionNumbers]{p}}+\sum_{k=1}^\infty \simplify[fractionNumbers]{1/k^{p}}\right)\end{align}$

$\begin{align}\sum_{k=\var{start}}^\infty \simplify[fractionNumbers]{{num}/({den}*k^{p})}&=\simplify[fractionNumbers]{{num}/{den}}\sum_{k=\var{start}}^\infty \simplify[fractionNumbers]{1/(k^{p})}\\&=\simplify[fractionNumbers]{{num}/{den}}\left(-1-\frac{1}{2^\var[fractionNumbers]{p}}-\frac{1}{3^\var[fractionNumbers]{p}}-\frac{1}{4^\var[fractionNumbers]{p}}-\frac{1}{5^\var[fractionNumbers]{p}}+\sum_{k=1}^\infty \simplify[fractionNumbers]{1/k^{p}}\right)\end{align}$

$\begin{align}\sum_{k=\var{start}}^\infty \simplify[fractionNumbers]{{num}/({den}*k^{p})}&=\simplify[fractionNumbers]{{num}/{den}}\sum_{k=\var{start}}^\infty \simplify[fractionNumbers]{1/(k^{p})}\\&=\simplify[fractionNumbers]{{num}/{den}}\left(-1-\frac{1}{2^\var[fractionNumbers]{p}}-\frac{1}{3^\var[fractionNumbers]{p}}-\frac{1}{4^\var[fractionNumbers]{p}}-\frac{1}{5^\var[fractionNumbers]{p}}-\frac{1}{6^\var[fractionNumbers]{p}}+\sum_{k=1}^\infty \simplify[fractionNumbers]{1/k^{p}}\right)\end{align}$

and so the convergence/divergence of our series depends on the convergence/divergence of the related $p$ -series. Note that adding or subtracting a finite number of terms to a series will not change whether it converges or diverges (such as our missing first ), nor will multiplying or dividing the series by a non-zero constant (such as $\frac{\var{num}}{\var{den}}$ ).

So we have a convergent series that you could say is really a $p$ -series with $p=\var[fractionNumbers]{p}$ .

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This question is used in the following exams:

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

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Taxonomy: mathcentre

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Taxonomy: Context

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Checkpoint description

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: `rational`

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Series: p series

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

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

Error

Warning

Generated value: rational

Parts

Gap-fill

Generated value: `rational`