Difference: ConvergenceTests (1 vs. 3)

Revision 32008-11-18 - Main.DickFurnas

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Convergence Tests

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Divergence Test

Statement
Changed:
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Comment
then may or may not converge.
Hint: Use the behavior of the limit, how goes to zero, as a clue!
>
>
Comment
then may or may not converge.
Hint: Ask yourself: How does go to zero? In the limit, does resemble a familiar sequence? Does the familiar series have known convergence properties? If so, you have the beginnings of a strategy for showing convergence or divergence.

P-Series

Statement

Comment

Geometric Series and related tests.

Statement

Comment
This is the granddaddy of many series which are easy to sum. It also is the foundation for several other tests when you observe:
  • The ratio of successive terms of the Geometric Series is x -- hence the Ratio Test
  • The ratio of the absolute values of successive terms of the Geometric Series is |x| -- hence the Ratio Test for Absolute Convergence
  • The nth root of the nth term of the Geometric Series is x -- hence the Root Test

Ratio Test

Statement

be a series with positive terms and

then:

Comment
Try this test when uk involves factorials or kth powers.

Ratio Test for Absolute Convergence

Statement

be a series with non-zero terms and

then:

Comment
The series need not have positive terms and need not be alternating to use this test since any series converges if it converges absolutely.

Root Test

Statement

be a series with positive terms and

then:

Comment: Try this test when uk involves kth powers.

Integral Test

Statement

be a series with positive terms and let f(x) be the function that results when k is replaced by x in the formula for uk.
If f is a decreasing, continuous function for x > N then:

have like convergence (either both converge or both diverge).

Comment
Use this test when f(x) is easy to integrate.

Limit Comparison Test

Statement
be series with positive terms such that:

then the two series have like convergence (either both converge or both diverge).
then notice which series "won".
  • Your unknown series converges if it is clearly smaller than a convergent series -- think about it.
  • Your unknown series diverges if it is clearly larger than a divergent series -- think about it.

Comment
This is easier to apply than the Comparison Test, but still requires some skill in choosing the known series. The Divergence Test can be a great source of inspiration here.
 

Comparison Test

Statement
be series with positive terms such that:

converges then converges.
Similarly, diverges, then diverges.
Changed:
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<
Comment
While this test is the foundation of most other tests, use it as a last resort. Other tests are often easier to apply.
>
>
Comment
Use this test as a last resort. While this test is the foundation of most other tests, other tests are often easier to apply.

Alternating Series Test

Statement

and
or equivalently
converges, provided
and
Comment
This test applies to alternating series only.

Telescoping Series

Statement
Any series where massive cancellation of terms occurs. Often partial sums simplify to a sum of some early terms and some ending terms: everything in between sums to zero (cancels).
Comment
Any time you see individual terms involving funky arithmetic with the indices, be on the lookout for a telescoping series.
  • Break up the typical term into a sum wherever possible.
  • Write out the first few terms.
  • Watch for developing patterns which will allow terms to cancel.

example... hide example

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Revision 22008-11-17 - Main.DickFurnas

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META TOPICPARENT name="InfiniteSeriesSynopsis"

Convergence Tests

Table of Contents
Changed:
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Name Statement Comments
<-- -->
Sorted ascending
Divergence Test If then may or may not converge.
Hint: Use the behavior of the limit, how goes to zero, as a clue!
Comparison Test Let be series with positive terms such that:

if converges then converges.
Similarly, if diverges, then diverges.
While this test is the foundation of most other tests, use it as a last resort. Other tests are often easier to apply.
>
>

Divergence Test

Statement
Comment
then may or may not converge.
Hint: Use the behavior of the limit, how goes to zero, as a clue!
 
Added:
>
>

Comparison Test

Statement
be series with positive terms such that:

converges then converges.
Similarly, diverges, then diverges.
Comment
While this test is the foundation of most other tests, use it as a last resort. Other tests are often easier to apply.
 
Added:
>
>

 -- DickFurnas - 16 Nov 2008

Revision 12008-11-16 - Main.DickFurnas

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Added:
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META TOPICPARENT name="InfiniteSeriesSynopsis"

Convergence Tests

Table of Contents

Name Statement Comments
Divergence Test If then may or may not converge.
Hint: Use the behavior of the limit, how goes to zero, as a clue!
Comparison Test Let be series with positive terms such that:

if converges then converges.
Similarly, if diverges, then diverges.
While this test is the foundation of most other tests, use it as a last resort. Other tests are often easier to apply.

-- DickFurnas - 16 Nov 2008

 
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