wk12 - MA104 Week 12 Power Series Concepts Absolute...

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Discrete Mathematics With Applications
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Chapter 4 / Exercise 32
Discrete Mathematics With Applications
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MA104 Week 12 ° Power SeriesConceptsAbsolute ConvergenceThe seriesPanis calledabsolutely convergentif the seriesPjanjis convergent and is calledconditionally convergentif it converges butPjanjdoes not.We then have the following theorem:IfPanis absolutely convergent, thenPanconverges.Note that the converse of the theorem is not true;that is, the convergence ofPandoes not implythatPanis absolutely convergent[ it may only be conditionally convergent instead ].Ratio and Root TestsTo determine if a series is absolutely convergent, the following tests may be helpful.Ratio Test: (i)Iflimn!1°°°°an+1an°°°°=L <1,then the series1Pn=1anis absolutely convergent.(ii)Iflimn!1°°°°an+1an°°°°=L >1orlimn!1°°°°an+1an°°°°=1,then the series1Pn=1anis divergent.Root Test: (i)Iflimn!1anis absolutely convergent.(ii)Iflimn!1npjanj=L >1orlimn!1npjanj=1,then the series1Pn=1anis divergent.In either test, if the limit evaluates toL= 1, no conclusion can be made as to the convergence ofPan.Also note that ifL= 1in the ratio test, then the root test will also result inL= 1.Rearrangement of an Infinite SeriesSuppose that we take an infinite seriesPanand create a rearrangement of the series by changingthe order of the terms in the original series.Then:(i)ifPanis absolutely convergent with sums,then any rearrangement ofPanalso has sums;(ii)ifPanis conditionally convergent andris any real number,then there existsa rearrangement ofPanthat has sumr.npjanj=L <1,then the series1Pn=1
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Discrete Mathematics With Applications
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Chapter 4 / Exercise 32
Discrete Mathematics With Applications
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