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 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.

example... hide example