What does a bindi represent

POSITIVESERIES:INTEGRALTEST,p-SERIES[SST8.3]THE SAD TRUTH ABOUT COMPUTING THE SUM OF A CONVERGENT SERIES:So far, we've seen two types of series whose sum can be determined (if convergent): geometric series & telescoping series.In general, it's very hard or impossible to determine the sum of a series by hand.Going forward, the focus will be ondetermining convergence using a collection of convergence tests.Later, we will nd sums of certain series usingTaylor series[SST 8.8].In higher math courses,Complex AnalysisandFourier Analysiscan be used to sum certain series.MORE SERIES NOTATION:In instances where the starting index doesn't matter, the series will be denoted byPak.This notation will be mostly used in thestatement of theorems & convergence tests.INSERTING/REMOVING FINITELY MANY TERMS DOES NOT ALTER CONVERGENCE OR DIVERGENCE:e.g.1Xk=0akconverges (diverges) =)1Xk=8akconverges (diverges) =)1Xk=17akconverges (diverges).e.g. WARNING: The inserted terms must be de ned: e.g.1Xk=31kis not well-de ned since the terma0=10is unde ned.POSITIVE SERIES:Pakis called apositive seriesif each termak08k.DIVERGENCE TEST:limk!1ak6= 0 =)Pakdiverges.TRANSLATION: "If the terms of the series do NOT converge to zero, then the series diverges."MEANING: Suppose limk!1ak=13. Then, "eventually" the seriesPakbecomes13+13+13+


which clearly diverges.So, the only hope for convergence is that the seriesPak"eventually" becomes 0 + 0 + 0 +


=)limk!1ak= 0WARNING: Just because limk!1ak= 0 does not necessarily mean that the seriesPakconverges.INTEGRAL TEST:Supposeak=f(k) fork=N;N+ 1;N+ 2;:::s.t.fiscontinuous&positive.Then:Z1Nf(x)dxconverges (diverges) =)positive series1Xk=Nakconverges (diverges).NOTE:Z1Nf(x)dx<1=)Z1Nf(x)dxconverges.Z1Nf(x)dx=1or DNE =)Z1Nf(x)dxdiverges.REMARK: Series involvingfactorials(e.g.k!) are disquali ed since theGamma Function() is too complicated.INTEGRAL DOMINANCE RULE:?(IDR)f;g2C[N;1) s.t.f(x)g(x)8x2[N;1) =)Z1Nf(x)dxZ1Ng(x)dx?Sometimes the initial integral is hard to evaluate, so using the Dominance Rule often leads to simpler integrals.?See the 8.3 Slides, 8.4 Slides, or the 8.4 Outline for a list of useful inequalities.p-SERIES TEST:p>1 =)p-series1Xk=11kpconverges.p1 =)p-series1Xk=11kpdiverges.c2013 Josh Engwer { Revised March 14, 2014
EX 8.3.1:Test the series1Xk=1ksin1kfor convergence.EX 8.3.2:Test the series1Xk=11ek+ekfor convergence.EX 8.3.3:Test the series1Xk=2lnk3pkfor convergence.EX 8.3.4:Test the series1Xk=120k2pk5for convergence.EX 8.3.5:Test the series1Xk=315k24pk3for convergence.EX 8.3.6:Test the series1Xk=11k13kfor convergence.c2013 Josh Engwer { Revised March 14, 2014