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 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 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 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.