Absolutely convergent series: Difference between revisions

Latest revision as of 11:59, 7 September 2011

A series of real numbers is termed an absolutely convergent series if it satisfies the following equivalent conditions:

The series obtained by taking absolute values of all terms in the series is a convergent series. Explicitly, a series of the form $\sum _{k=1}^{\infty }a_{k}=a_{1}+a_{2}+a_{3}+\dots$ is absolutely convergent if $\sum _{k=1}^{\infty }|a_{k}|=|a_{1}|+|a_{2}|+|a_{3}|+\dots$ is a convergent series.
The sub-series of the series comprising the terms with positive values is convergent, and the sub-series of the series comprising the terms with negative values is also convergent.
Every rearrangement of the series is a convergent series and converges to the same value as the series itself.

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 A [[series]] of real numbers is termed an '''absolutely convergent series''' if it satisfies the following equivalent conditions:
-# The series obtained by taking absolute values of all terms in the series is a [[convergent series]]. Explicitly, a series of the form <math>\sum_{k=1}^infty a_k = a_1 + a_2 + a_3 + \dots</math> is absolutely convergent if <math>\sum_{k=1}^\infty |a_k| = |a_1| + |a_2| + |a_3| + \dots</math> is a convergent series.
+# The series obtained by taking absolute values of all terms in the series is a [[convergent series]]. Explicitly, a series of the form <math>\sum_{k=1}^\infty a_k = a_1 + a_2 + a_3 + \dots</math> is absolutely convergent if <math>\sum_{k=1}^\infty |a_k| = |a_1| + |a_2| + |a_3| + \dots</math> is a convergent series.
+# The sub-series of the series comprising the terms with positive values is convergent, and the sub-series of the series comprising the terms with negative values is also convergent.
 # Every rearrangement of the series is a convergent series and converges to the same value as the series itself.