Dr. Therese Shelton

Associate Professor
Department of Mathematics and Computer Science
Southwestern University
Georgetown, TX


Summary of Convergence and Divergence Tests

Test Series Convergence or Divergence Comments
Ratio If , the series
  1. converges absolutely if L < 1
  2. diverges if L > 1 or is infinite
Inconclusive otherwise.
Useful for factorials and nth powers.
For power series, let an be the entire power series term.
nth-term Diverges if Inconclusive if the limit is zero.
geometric series
  1. Converges with sum if |r|<1
  2. Diverges otherwise
Useful for comparison tests
p-series
  1. Converges if p > 1
  2. Diverges otherwise
Useful for comparison tests
Integral
f(n) = an,
f(x) defined for x>1 is continuous, positive, and decreasing
(check f'(x) )
  1. Converges if converges
  2. Diverges otherwise
 
Basic Comparison given,
usually is chosen,
all terms in both series strictly positive
  1. If converges and an <= bn for every n, then converges
  2. If diverges and an >= bn for every n, then diverges
Inconclusive otherwise.
Often the b-series is geometric or a p-series. Consider terms of the a-series of greatest effect.
Limit Comparison given,
usually is chosen,
all terms in both series strictly positive
If is finite then both series converge or both diverge. Inconclusive otherwise.
Often the b-series is geometric or a p-series. Consider terms of the a-series of greatest effect.
Root If , the series
  1. converges absolutely if L < 1
  2. diverges if L > 1 or is infinite
Inconclusive otherwise.
Useful for nth powers, such as in power series.
Absolute series If , then converges Inconclusive otherwise.
Useful for series with both positive and negative terms.
Alternating series , an > 0 Converges if for every k and Inconclusive otherwise.
Note that an is the non-alternating part!
If inequalities are not easy, let f(n) = an.
f(x) defined for x>1 should be continuous, positive, and decreasing
(check f'(x) )

Common Series


and their intervals of convergence

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Created : Nov 26, 2001
Last modified : Nov 26, 2001
Author : Therese Shelton email : shelton@southwestern.edu


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