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Series and Final Exam ReviewApril 13, 20181Convergence & Divergence Tests1.1Divergence TestIf limn→∞an6= 0, thenanwill diverge.This testONLYsays that a series is guaranteed to diverge if the seriesterms don’t go to zero in the limit.This test does not guarantee that the series diverges if the series terms goto zero in the limit.1.2Geometric SeriesA geometric series has the formn=1arn-1The partial sums of this type of series aresn=a(1-rn)1-r=a1-r-arn1-rThis series converges provided the partial sums form a convergent se-quence.A geometric series will converge if|r|<1, its value isa1-r1.3P-SeriesAp-serieshas the formn=k1npand will converege ifp >1.Note that a special case of thep-seriesis whenp= 1. This is known astheharmonic seriesand diverges.1
1.4Alternating Series TestSuppose that we have a seriesanand eitheran= (-1)nbnoran= (-1)n+1bnwherebn0 for alln. Then if,limn→∞bn= 0 andbnis a decreasing sequenceThe seriesanwill be convergent.This test only tells us when a series is convergent.Secondly, in the second condition all that we require is that the termsbnwill be eventually decreasing.note that a special case of the alternating series is theAlternating Har-monic Seriesand has the formn=1(-1)n+1n1.5Absolute ConvergenceA seriesanis calledabsolutely convergentif|an|is convergent. Ifanisconvergent and|an|is divergent we call the seriesconditionally convergent.Ifanis absolutely convergent then it is also convergent.1.6Ratio TestSuppose we have the seriesanDefine,L= limn→∞an+1anThen,IfL <1 Then the series is absolutely convergent (and hence convergent).IfL >1 Then the series is divergent.IfL= 1 Then the series may be divergent, conditionally convergent, orabsolutely convergent.Note that applying the ratio test to a rational expression involving onlypolynomials or polynomials under radicals will always result inL= 1 soit’s not worth using this test for those cases.2
1.7Root Test

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