Determineif the series convergesabsolutely.
Solution
The seriesmadeup of the absolutevaluesof the termsis This
seriesbehaveslike , whichdiverges.The series does not converge
absolutely.
It is possibleto havea seriesthat is convergent,but not a absolutelyconvergent.
ConditionalConvergenceAn infiniteseriesthat converges,but doesnot convergeabsolutely, is
calledaconditionallyconvergentseries.Example 5
Determineif convergesabsolutely, convergesconditionally, or diverges.
Solution
The seriesof absolutevaluesis. This is the harmonicseries,whichdoesnot converge.So, the series
doesnot convergeabsolutely. The next step is to checkthe convergence.
This will tell us if the seriesconvergesconditionally. Applyingthe AlternatingSeriesTest:
The sequence is nondecreasingand.
The series converges.Hence,the seriesconvergesconditionally, but not absolutely.
Rearrangement
Makingarearrangementof termsof a seriesmeanswritingall of the termsof a seriesin a differentorder.
The followingtheoremexplainshow rearrangementaffectsconvergence.
TheoremIf is an absolutelyconvergentseries,then the new seriesformedby a rearrangement
of the termsof the seriesalso convergesabsolutely.