Math 150B Exam 2 Solution

Exam (2)Math 150B. Calculus IISection # 18056 Fall2022Instructor: Henrik SarkssianMoruoay OcroeEn 24, 7 2:3O PM - I :35 PMName:- ShOW(Please print)all Of yOUf WOfk and make the final answers clear. Full credit will NoT be givenfor merely stating the right answer without any appropriate justification.- Cell phones must not be visible or audible at any time during the exam. This applies to all smarttechnology including watches, cameras, and pens.NoCa lculators!Your qrade witl be pos+ed on Canvas ovce yot subw\t a,opy bf +lAo qradod +es+ +lAore (no rodoes, wo clnawqos).yot fat to subwl+, yot will c)o+ a Lero.ffDus Datoto submit this tostot/1canVas:Problem1234TotalMaximumt30l[30]t30l[60]150Score1Exam (2) Math 150B. Calculus II Section # 18056 Fall 2022 Instructor: Henrik Sarkssian MONDAY OCTOBER 24, 12:30 PM - 1:35 PM Name: Answer key (Please print) - Show all of your work and make the final answers clear. Full credit will NOT be given for merely stating the right answer without any appropriate justification. - Cell phones must not be visible or audible at any time during the exam. This applies to all smart technology including watches, cameras, and pens. No Calculators! Your grade will be posted on Canvas once you submit a copy of the graded test there (no redoes, no changes). If you fail to submit, you will get a zero. Due Date to submit this test on canvas: 11/03/22 Problem 1 2 3 4 Total Maximum [30] [30] [30] [60] 150 Score 1

Problem 1. (30 pts.)Determine if the integral converges or diverges.First express the improper integral as a limit. De[']*I,..ILrtv=*,\lb"t;:)o J+-tfitro^ = Q.*n [os Au 8*'kb; z"- !,u'( o) -&S=8u *ieilLt0b.9.E -O 5 7aac 0dt^\vYd, ,r^ktJCAoil,vyof,Problem 1. (30 pts.) First express the improper integral as a limit. Determine if the integral converges or diverges. 8 b = lim shadd = lim b 10 b+2 4-x 2 dinns and 0 b-12 M988: I lim Sin't - snco) I-O-I 2 2 The integral converges hiz y 3 y y nevis ed TON lliw tibato 1109 Isnit erls noitspiliteu tuodtiw sigh arlj visism lls of esilqqs eidT art gnhub smit yos 16 eldibus 10 eldiziv ad ton teum llso bns 2679M83 291016W ygolonrbef s timidine NON gd liw sbore YNOP 7I ON test bobaye SNT to NJOU lliw NON timona of list NON test zint timona of STAT IstoT S & meldor9 021 [0a] [08] [0E] (OE) mumixsM 91032 2

Problem 2. (30 pts.)Evaluate the limit of each sequence. Show all of vour work.,.\(t)(n-l)"tt I\"(l==f ^/\n )?n=l|fur''( /- *).&u(/\t@+g*i--Lv^X-+ xsur''(-.a)''r ****,'tl"t'ttv O,!rwx'too+,-l4-l^x'r;l)tr-e0w;I'-+w'XfrA-o-\r-^-r:r Lt:'- I{'-/o3Problem 2. (30 pts.) Evaluate the limit of each sequence. Show all of your work. (i) allow ) will fix1s(1-1) P.O.N x(n(1-4) lim Cim (1-2)* lim e X-00 KID X-00 3 s e X-00 2 of L'Hop 2 s e 1/2 -1 lim 1-x lim 1-1/2020 e X-00 -1/2 X-00 +1 se s e 3. 3 serve 3 3

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Problem 3. (30 pts.)Find the exact sum of the series.(i) Ic-fu=tZnE(n+rr(kfr)+r|('&rn-l !,\ 4=Lij 'lh[o -u(*,).Tvt_eln46+ iL(}-])"l[G-*)nsj^st' fit1Tl= *(* *:+)",^( +-(-,_,i I(+-.) + i( +-,)j)t1r6?j_tL 6,!: *)l-r L ,-rtr-t- tisn-Problem 3. (30 pts.) HOT Find the exact sum of the series. so 00 00 (i) n=3 = 1/2 ns3 (n+1)-(n-1) (n-1)(n+1) 1/1 n-1 - ne! 1/1 ns3 is 13/14 - 1/2 2 n-1 1-1-1/1 - 1 is = nel no3 Do 1/2 51/30 2 nez ml + 1/2 "N (1/2) n+1 n=3 ns3 3 3 = 1/2 (st-lim 7) + 1/6/1/20 lim 11 noo ke! E 2 2 = 14+1/16 = 5 12 2

(ii)I(+)($)vr(?)^lltlI3t-3L;'l'IZE,Z,-L3/6-)1,l-,L.5j)),-,G;=/-fr))1esl_l6,6,\/lfl.'(ii) (5) n=1 - 2" 5" (.arq 0E) .£ meldor9 to bail 52 (1) - (7) NY 3 ns1 1/3/k1 G/KI 3 1/2 5 3 2/1 5 1- 3 2 3 5 3 5 3 2 5 2 3 3 5 582 2 3 = 11 6 (ont 3 (0-1) a s SI 6

Problem 4. (60 pts.)By using an appropriate test (specifu the name determine if the series converge absolutely,converge conditionally, or diverge.(i)I# y r*"hnl;D?.wT'Y-^-pYL(ii)I#'*,h','* n'€-v",9*r\4V00r\t2P-senLrt>2 bM/ryor?|-pnV,r@ttuerlery*t-f ,,tf* |J, xht-dr:/pc,.o^A.6@b*'u-Gx(^@pClLa)(o

I1t-oofrl#l'If+4v q(iii), -L,^\;tn-,,FA)Lv,UooL,C,r)fL.f?nI/)31/.4N-/ L -LAl4s-lrLH^rn*r'.l);'re4 lim Mos 3th D 3 1st OLLC 00 By A.S.T bns 1/2 3nol 7 2 (i) lim 1 50 200 3ne! (ii) fixis 1 3x4' fixis (3x+1)2 - -3 2 nole 3 1 / now n MIX The series diverger by Ratis test. G