Difference: ConvergenceTests (3 vs. 4)

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Convergence Tests

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Divergence Test

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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

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Geometric Series and related tests.

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This is the granddaddy of many series which are easy to sum. It also is the foundation for several other tests when you observe:
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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
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    • The nth root of the nth term of the Geometric Series is x -- hence the Root Test
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    • The nth root of the nth term of the Geometric Series is x -- hence the Root Test
 

Ratio Test

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  be a series with positive terms and

then:
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Try this test when uk involves factorials or kth powers.
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Try this test when uk involves factorials or kth powers.
 

Ratio Test for Absolute Convergence

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  be a series with non-zero terms and

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

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  be a series with positive terms and

then:
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Comment: Try this test when uk involves kth powers.
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Comment: Try this test when uk involves kth powers.
 

Integral Test

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  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.
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If f is a decreasing, continuous function for x > N then:
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If f is a decreasing, continuous function for x > N then:
 
have like convergence (either both converge or both diverge).
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Use this test when f(x) is easy to integrate.
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Use this test when f(x) is easy to integrate.
 

Limit Comparison Test

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be series with positive terms such that:
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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.
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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.
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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

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be series with positive terms such that:
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be series with positive terms such that:
 
converges then converges.
Similarly, diverges, then diverges.
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Use this test as a last resort. While this test is the foundation of most other tests, other tests are often easier to apply.
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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

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

Telescoping Series

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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).
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Any time you see individual terms involving funky arithmetic with the indices, be on the lookout for a telescoping series.
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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).
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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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 -- DickFurnas - 16 Nov 2008
 
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