
META TOPICPARENT 
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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.


PSeries 

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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 n^{th} root of the n^{th} term of the Geometric Series is x  hence the Root Test

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 The n^{th} root of the n^{th} 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 u_{k} involves factorials or k^{th} powers.

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 Try this test when u_{k} involves factorials or k^{th} powers.


Ratio Test for Absolute Convergence 

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 be a series with nonzero 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:


< <  Comment: Try this test when u_{k} involves k^{th} powers. 
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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 u_{k}. 

< <  If f is a decreasing, continuous function for x > N then: 
> >  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).
 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.
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  DickFurnas  16 Nov 2008 