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Series Summary


Divergence or nth term test
Series:
Convergence or Divergence: Diverges if
Comments: Nothing can be said if


Geometric Series Test
Series:
Convergence or Divergence:
* converge and have sum if | r | <1.

* diverges if | r | 1

Comments: Useful for comparison tests


P-series test
Series:
Convergence or Divergence:
* converge if p> 1
* diverges if p 1

Comments: Useful for comparison tests


Limit comparison test
Series: and , ano > 0, bno > 0

Convergence or Divergence:
* If , , then both series converge or both diverge.

* If and converge then converge.

* If and diverges then diverges.

Comments: The Comparison Series , is usually a geometric series or a p-series.

To find bno, only the terms of theno that have the greatest effect.


Leibniz test
Series: ALTERNATE
, ano > 0
Convergence or Divergence:
Converge if:

*

* The series of the modules is decreasing.

Comments: Applicable only to alternating series. If the first item is false, the divergence test applies.