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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.
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P-Series |
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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:
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- 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 nth root of the nth term of the Geometric Series is x -- hence the Root Test
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- The nth root of the nth term of the Geometric Series is x -- hence the Root Test
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Ratio Test |
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| | be a series with positive terms and
then:
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- Try this test when uk involves factorials or kth powers.
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- Try this test when uk involves factorials or kth powers.
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Ratio Test for Absolute Convergence |
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| | be a series with non-zero 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.
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Root Test |
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| | be a series with positive terms and
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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 uk.
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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.
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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:
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 then the two series have like convergence (either both converge or both diverge).
 then notice which series "won".
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- 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.
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Comparison Test |
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- be series with positive terms such that:
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- be series with positive terms such that:
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 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.
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Alternating Series Test |
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or equivalently 
converges, provided 
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- This test applies to alternating series only.
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- This test applies to alternating series only.
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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).
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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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- 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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