# HW12-solutions - garcia(pgg378 HW12 he(53725 This print-out...

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garcia (pgg378) – HW12 – he – (53725)1Thisprint-outshouldhave24questions.Multiple-choice questions may continue onthe next column or page – find all choicesbefore answering.00110.0pointsCompare the values of the seriesA=summationdisplayn=15n1.7and the improper integralB=integraldisplay15x1.7dx .1.A > Bcorrect2.A=B3.A < BExplanation:In the figure12345. . .a1a2a3a4the bold line is the graph of the functionf(x) =5x1.7on [1,) and the area of each of the rectan-gles is one of the values ofan=5n1.7.Clearly from this figure we see thata1=f(1)>integraldisplay21f(x)dx,a2=f(2)>integraldisplay32f(x)dx ,whilea3=f(3)>integraldisplay43f(x)dx,a4=f(4)>integraldisplay54f(x)dx ,and so on for alln. Consequently,A=summationdisplayn=15n1.7>integraldisplay15x1.7dx=B.00210.0pointsWhich of the following series are convergent:A.1 +14+19+116+. . .B.summationdisplayn=13n+ 1C.summationdisplayn=12n2/31.A and C only2.B only3.C only4.A onlycorrect5.A and B only6.none of them7.B and C only
garcia (pgg378) – HW12 – he – (53725)28.all of themExplanation:By the Integral test, iff(x) is a positive,decreasing function, then the infinite seriessummationdisplayn=1f(n)converges if and only if the improper integralintegraldisplay1f(x)dxconverges. Thus for the three given series wehave to use an appropriate choice off.A. Usef(x) =1x2. Thenintegraldisplay1f(x)dxis convergent.B. Usef(x) =3x+ 1. Thenintegraldisplay1f(x)dxis divergent (log integral).C. Usef(x) =2x2/3. Thenintegraldisplay1f(x)dxis divergent.keywords: convergent, Integral test,00310.0pointsWhich, if any, of the following series con-verge?A.1 +14+19+116+. . .B.summationdisplayn=111 +n21.A only2.neither of them3.B only4.both of themcorrectExplanation:A. Series issummationdisplayn=11n2. Usef(x) =1x2. Thenintegraldisplay1f(x)dxis convergent, so series converges.B. Usef(x) =11 +x2. Thenintegraldisplay1f(x)dxis convergent (tan1integral), so series con-verges.00410.0pointsFirst findanso thatsummationdisplayn=1an= 6 +32+23+34+655+. . .and then determine whether the series con-verges or diverges.1.an=6n1/2,series diverges2.an=32n3/2,series diverges3.an=6n1/2,series converges4.an=6n3/2,series diverges5.an=6n3/2,series convergescorrect