L21 Absolute Convergence Ratio and Root Tests Part I - 1 If lim n!1 j a n 1 a n j = L < 1 then the series is absolutely convergent 2 If lim n!1 j a n 1

# L21 Absolute Convergence Ratio and Root Tests Part I - 1 If...

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Lecture 21: Absolute Convergence andRatio and Root TestsConsider the two convergent alternating series(a)X(1)n+1nand(b)X(1)n+1n2Consider the series whose terms are the absolutevalues of the terms of the original series in (a)and (b):X1nandX1n2SinceX1ndiverges and the orginal series (a)converges, we say that series (a) iscondition-allyconvergent.SinceX1n2converges, we say that series (b)isabsolutely convergent.1
Basically the question is:DoesPanconverges or diverges?GivenXan, consider the seriesXjanj:Def.IfXjanjconverges, then we callXanabsolutely convergent.IfXjanjdiverges, butXanconverges,then we callXanconditionaly convergentTheorem:If a seriesXanis absolutelyconvergent, then it is convergent.2
ex.Xcosnn23
Ratio Test-LetXanbe a series.