Lecture 19:Comparison TestGeneral Idea: Compare the two series1Xn=113n+ 2and1Xn=113nThe second series converges (geometric) and13n+ 2<13nHence, therst series must converges, too.Theorem:Suppose thatPanandPbnareseries with positiveterms.(1) IfPbnis convergent andanbn;thenPanis convergent.(2) IfPbnis divergent andanbn;thenPanis divergent.Note:Ifanbn;and the seriesPbnis di-vergent, no conclusion can be drawn about theseriesPan.1

Examples:(1)1Xn=113pn1(2)1Xn=3lnnn(3)1Xn=01n2+ 52

The Limit Comparison TestIdea: The Comparison test is not convenient forthe series1Xn=113pn+ 1But, you can use theLimit Comparison Test:Theorem:Suppose thatPanandPbnareseries with positive terms. Iflimn!1anbn=cwherec >0 is a