Convergence Tests
Divergence Test
 Statement

 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.
PSeries
 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 n^{th} root of the n^{th} 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 u_{k} involves factorials or k^{th} powers.
Ratio Test for Absolute Convergence
 Statement

be a series with nonzero 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 u_{k} involves k^{th} 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 u_{k}.
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.
 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

DickFurnas  16 Nov 2008