Series: divergence test

Error

There was an error loading the page.

Contributors

Ben Brawn

Feedback

From users who are members of Sequences and Series :

Ben Brawn

said

Ready to use

7 years, 8 months ago

History

○

Ben Brawn 7 years, 8 months ago

Published this.

○

Ben Brawn 7 years, 8 months ago

Gave some feedback: Ready to use

○

Ben Brawn 7 years, 8 months ago

Published this.

○

Ben Brawn 7 years, 8 months ago

Created this.

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

There are 4 other versions that do you not have access to.

Question statement

This question is about the divergence test for series.

Give any introductory information the student needs.

			Name	Type	Generated Value

text	string	larger
seed	integer	1
limit	integer	0
existence	integer	0
cseed	string	inf
a	integer	3
d	integer	7
coeff	list	[ -9, -12, -3, 3, 10 ]
b	number	-9
c	number	-12
f	number	-3
g	number	3
h	number	10
alt	integer	0
cexpression	string	$\displaystyle\simplify{{a}^k}
con_to_zero	list	List of 7 items
div_to_inf	list	List of 7 items
jump_around	list	List of 12 items
con_to_non_zero	list	List of 5 items
start	integer	3

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
if(seed=1,'larger', 'smaller')

JME function reference

Description

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

Can an exam override the value of this variable?

Generated value: `string`

larger

← Depends on:

seed

Choose one from a list

Ask the student a question, and give any hints about how they should answer this part.

A series is such that each term has an absolute value that is {text} than the last. What does the divergence test tell us about this series?

Show choice feedback state?

Show correct answer on reveal?

Show score feedback icon?

Advanced settings

Selection type

Custom marking matrix?

This series converges.

Marks: Distractor Message:
This series diverges.

Marks: Distractor Message:
It doesn't tell us anything.

Marks: Distractor Message:

Advice

The divergence test says that if the sequence of $t_k$ diverges or converges to a non-zero number then the series $\sum_{k=a}^\infty t_k$ diverges. Notice it does not tell us anything about the series if the sequence of $t_k$ converges to $0$ . Another way to think about this is, for a series to have any chance of converging then we require its terms to converge to zero. You might see the divergence test written more concisely as

$\lim_{k\rightarrow\infty} t_k\ne 0 \implies \sum_{k=0}^\infty t_k \,\,\text{ diverges.}$

Since each term in the series is getting {text} in absolute value the sequence can never 'settle down' or 'approach' a finite number. In other words, the sequence diverges (does not converge) and so the divergence test implies that the series diverges.

Since each term in the series is getting {text} in absolute value the sequence 'settles down' and 'approaches' zero. In other words, the sequence converges to zero and so the divergence test doesn't actually tell us anything about the series (it may or may not converge).

Since the limit of the sequence does not exist, the sequence diverges (diverges just means does not converge) and so the divergence test tells us that the series diverges.

Since the limit of the sequence is a non-zero number ( $\var[fractionNumbers]{limit}$ ) the divergence test tells us that the series diverges.

Since the limit of the sequence is zero, the divergence test doesn't actually tell us anything about whether the series converges or diverges.

To use the divergence test we need to determine whether the sequence of terms approach zero or not. Consider

{cexpression} as $k\rightarrow \infty$ .

As $k$ gets larger and larger our terms get larger and larger. This means that the limit does not exist (or sometimes people prefer to say the limit equals infinity) and so the divergence test tells us that the series diverges.

As $k$ gets larger and larger our terms get larger and larger in absolute value but alternate in sign. This means that the limit does not exist and so the divergence test tells us that the series diverges.

As $k$ gets larger and larger our terms jump around, they get positive, negative, smaller, larger... The terms do not approach a constant. This means that the limit does not exist and so the divergence test tells us that the series diverges.

As $k$ gets larger and larger our terms approach zero. The larger $k$ gets, the closer to zero the terms become. This means that the sequence converges to $0$ and so the divergence test does not tell us anything about whether the series converges or diverges.

As $k$ gets larger and larger our terms approach a non-zero constant. The larger $k$ gets, the closer to this non-zero constant the terms become. This means that the sequence converges to this non-zero constant and so the divergence test tells us that the series diverges.

Give a worked solution to the whole question.

Use this tab to check that this question works as expected.

There was an error which means the tests can't run:

Part	Test	Passed?
Choose one from a list
Choose one from a list		Hasn't run yet
Choose one from a list
Choose one from a list		Hasn't run yet
Choose one from a list
Choose one from a list		Hasn't run yet

This question is used in the following exams:

Series by Ben Brawn in Sequences and Series.
analysis by Daniel Atcheson in Daniel's workspace.

Loading...

Error

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Ben Brawn 7 years, 8 months ago

Ben Brawn 7 years, 8 months ago

Ben Brawn 7 years, 8 months ago

Ben Brawn 7 years, 8 months ago

Error in variable testing condition

Error

Warning

Generated value: `string`

Parts

Choose one from a list

Advanced settings

Loading...

Error

Series: divergence test

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Ben Brawn 7 years, 8 months ago

Ben Brawn 7 years, 8 months ago

Ben Brawn 7 years, 8 months ago

Ben Brawn 7 years, 8 months ago

Error in variable testing condition

Error

Warning

Generated value: string

Parts

Choose one from a list

Advanced settings

Generated value: `string`