Series: divergence test

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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 divergence test for series.

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			Name	Type	Generated Value

text	string	smaller
seed	integer	-1
limit	rational	-1/3
existence	integer	0
cseed	string	non_zero
a	integer	5
d	integer	-7
coeff	list	[ -4, 0, 0, 0, -7 ]
b	number	-4
c	integer	0
f	integer	0
g	integer	0
h	number	-7
alt	integer	1
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 7 items
start	integer	1

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

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smaller

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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?

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This series converges.

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This series diverges.

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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.

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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.

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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

Ben Brawn 7 years, 7 months ago

Error in variable testing condition

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Series: divergence test

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

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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

Ben Brawn 7 years, 7 months ago

Error in variable testing condition

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Generated value: `string`