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.