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Absolute Convergence
Absolutely Convergent

Describes a series that converges when all terms are replaced by their absolute values. To see if a series converges absolutely, replace any subtraction in the series with addition. If the new series converges, then the original series converges absolutely.

Note: Any series that converges absolutely is itself convergent.

 

Definition:    A series is absolutely convergent if the series converges.
   
Example:

Determine if    is absolutely convergent.

   
Solution:

To find out, consider the series   .

 

This is an infinite geometric series with ratio , so it converges to or 2. As a result, we know that converges absolutely.

 

See also

Convergence tests

 


  this page updated 21-feb-16
Mathwords: Terms and Formulas from Algebra I to Calculus
written, illustrated, and webmastered by Bruce Simmons
NCTM Web Bytes December 2004 Web Bytes March 2005 Web Bytes