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.
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.
- 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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DickFurnas - 16 Nov 2008