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Absolute And Conditional Convergence - Definition

9.6 ABSOLUTE AND CONDITIONAL CONVERGENCE

Consider a series

09_infinite_series-313.gif

which has both positive and negative terms. We may form a new series

09_infinite_series-314.gif

whose terms are the absolute values of the terms of the given series. If all the terms an are nonzero, then |an| > 0 so

09_infinite_series-315.gif

is a positive term series.

If 09_infinite_series-316.gif is already a positive term series, then |an| = an and the series is identical to its absolute value series 09_infinite_series-317.gif

Sometimes it is simpler to study the convergence of the absolute value series 09_infinite_series-318.gif than of the given series 09_infinite_series-319.gif. This is because we have at our disposal all the convergence tests for positive term series from the preceding sections.

DEFINITION

A series 09_infinite_series-320.gif is said to be absolutely convergent if its absolute value series 09_infinite_series-321.gif is convergent. A series which is convergent but not absolutely convergent is called conditionally convergent.
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Last Update: 2006-11-07