to put into appropriate form NO Does lim 64257nite YES YES Diverges NO TAYLOR SERIES Does NO Is in interval of convergence YES 0 YES Diverges NO Try one or more of the following tests NO COMPARISON TEST Pick Does converge Is 0 YES Converges YES Is ID: 43694
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SeriesConvergence/DivergenceFlowChartTESTFORDIVERGENCE Doeslimn!1an=0? PanDiverges NOp-SERIES Doesan=1=np,n1? YES Isp�1? YES PanConverges YES PanDiverges NOGEOMETRICSERIES Doesan=arn 1,n1? NO Isjrj1? YES P1n=1an=a 1 r YES PanDiverges NOALTERNATINGSERIES Doesan=( 1)nbnoran=( 1)n 1bn,bn0? NO Isbn+1bn&limn!1bn=0? YES PanConverges YESTELESCOPINGSERIES Dosubsequenttermscanceloutprevioustermsinthesum?Mayhavetousepartialfractions,propertiesoflogarithms,etc.toputintoappropriateform. NO Doeslimn!1sn=ss nite? YES Pan=s YES PanDiverges NOTAYLORSERIES Doesan=f(n)(a) n!(x a)n? NO Isxinintervalofconvergence? YES P1n=0an=f(x) YES PanDiverges NO Tryoneormoreofthefollowingtests: NOCOMPARISONTEST Pickfbng.DoesPbnconverge? Is0anbn? YES PanConverges YES Is0bnan? NO NO PanDiverges YESLIMITCOMPARISONTEST Pickfbng.Doeslimn!1an bn=c�0c nite&an;bn�0? Does1Xn=1bnconverge? YES PanConverges YES PanDiverges NOINTEGRALTEST Doesan=f(n),f(x)iscontin-uous,positive&decreasingon[a;1)? DoesZ1af(x)dxconverge? YES P1n=aanConverges YES PanDiverges NORATIOTEST Islimn!1jan+1=anj6=1? Islimn!1 an+1 an 1? YES PanAbs.Conv. YES PanDiverges NOROOTTEST Islimn!1np janj=1? Islimn!1np janj1? YES PanAbs.Conv. YES PanDiverges NO Problems1-38fromStewart'sCalculus,page7841.1Xn=1n2 1 n2+n2.1Xn=1n 1 n2+n3.1Xn=11 n2+n4.1Xn=1( 1)n 1n 1 n2+n5.1Xn=1( 3)n+1 23n6.1Xn=13n 1+8nn7.1Xn=21 np ln(n)8.1Xk=12kk! (k+2)!9.1Xk=1k2e k10.1Xn=1n2e n311.1Xn=2( 1)n+1 nln(n)12.1Xn=1( 1)nn n2+2513.1Xn=13nn2 n!14.1Xn=1sin(n)15.1Xn=0n! 258(3n+2)16.1Xn=1n2+1 n3+117.1Xn=1( 1)n21=n18.1Xn=2( 1)n 1 p n 119.1Xn=1( 1)nln(n) p n20.1Xk=1k+5 5k21.1Xn=1( 2)2n nn22.1Xn=1p n2 1 n3+2n2+523.1Xn=1tan(1=n)24.1Xn=1cos(n=2) n2+4n25.1Xn=1n! en226.1Xn=1n2+1 5n27.1Xk=1kln(k) (k+1)328.1Xn=1e1=n n229.1Xn=1tan 1(n) np n30.1Xj=1( 1)jp j j+531.1Xk=15k 3k+4k32.1Xn=1(2n)n n2n33.1Xn=1sin(1=n) p n34.1Xn=11 n+ncos2(n)35.1Xn=1n n+1n236.1Xn=21 (ln(n))ln(n)37.1Xn=1(np 2 1)n38.1Xn=1(np 2 1)