Spring 2016 Exam 2 NAME: PIN:

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1 MARK BOX problem poits NAME: PIN: % 00 INSTRUCTIONS O Problem 0, fill i the blaks. As you kow, if you do ot make at least half of the poits o Problem 0, the your score for the etire exam will be whatever you made o Problem 0. For multiple choice problems, circle your aswer(s) o the provided chart. No eed to show work. For all other problems, to receive credit you MUST show ALL your work ad : () work i a logical fashio, show all your work, idicate your reasoig; o credit will be give for a aswer that just appears; such explaatios help with partial credit (2) if a lie/box is provided, the: show you work BELOW the lie/box put your aswer o/i the lie/box (3) if o such lie/box is provided, the box your aswer. Upo request, you will be give as much (blak) scratch paper as you eed. Check that your copy of the exam has all of the problems. Durig the exam, the use of uauthorized materials is prohibited. Uauthorized materials iclude: electroic devices, books, ad persoal otes. Uauthorized materials (icludig cell phoes) must be i a secured (e.g. zipped up, sapped closed) bag placed completely uder your desk or, if you did ot brig such a bag, give to Prof. Girardi to hold for you durig the exam (ad they will be retured whe you leave the exam). This meas o electroic devices (such as cell phoes) allowed i your pockets. At a studets request, I will project my watch upo the projector scree. Durig this exam, do ot leave your seat uless you have permissio. If you have a questio, raise your had. Whe you fiish: put your pecil dow ad raise your had. This exam covers (from Calculus by Thomas, 3 th ed., ET): Hoor Code Statemet I uderstad that it is the resposibility of every member of the Carolia commuity to uphold ad maitai the Uiversity of South Carolia s Hoor Code. As a Caroliia, I certify that I have either give or received uauthorized aid o this exam. I uderstad that if it is determied that I used ay uauthorized assistace or otherwise violated the Uiversity s Hoor Code the I will receive a failig grade for this course ad be referred to the academic Dea ad the Office of Academic Itegrity for additioal discipliary actios. Furthermore, I have ot oly read but will also follow the istructios o the exam. Sigature : Prof. Girardi Page of 9 Math 42

2 0. Fill-i-the boxes. All series are uderstood to be, uless otherwise idicated. 0.. Geometric Series. Fill i the boxes with the proper rage of r R. The series r coverges if ad oly if r satisfies p-series. Fill i the boxes with the proper rage of p R. The series p coverges if ad oly if State the Itegral Test for a positive-termed series a. Let f : [, ) R be so that a = f () for each N f is a f is a f is a The a coverges if ad oly if fuctio fuctio fuctio. coverges State the Compariso Test for a positive-termed series a. Let N 0 N (e.g., N 0 might be 7). If whe N 0 ad, the a coverges. If whe N 0 ad, the a diverges. Hit: sig the sog to yourself State the Limit Compariso Test for a positive-termed series a. Let b > 0 ad L = lim a b. If 0 < L <, the. Goal: cleverly pick positive b s so that you kow what b does (coverges or diverges) ad the sequece { a b } coverges By defiitio, for a arbitrary series a, (fill i these 3 boxes with coverget or diverget). a is absolutely coverget if ad oly if a is. a is coditioally coverget if ad oly if a is ad a is. a is diverget if ad oly if a is diverget State the Ratio ad Root Tests for arbitrary-termed series a with < a <. Let ρ = lim a + a or ρ = lim a. If If If the a coverges absolutely. the a diverges. the the test is icoclusive State the Alteratig Series Test (AST). If () u > 0 for each N (2) lim u = (3) u u + for each N, the Page 2 of 9

3 . Circle T if the statemet is TRUE. Circle F if the statemet if FALSE. To be more specific: circle T if the statemet is always true ad circle F if the statemet is NOT always true. Scorig: 2 pts for correct aswer, 0 pts for a icorrect aswer, pt for a blak aswer (idicated by a circled B). O the ext 3, thik of the th -term test for divergece ad what if a = T F B If lim a 0, the a diverges. T F B If a coverges, the lim a = 0. T F B If lim a = 0, the a coverges. O the ext 5, thik of AC vs. CC vs. Diverget. Examples from Problem 2 might be helpful. T F B A series a is precisely oe of the followig: absolutely coverget, coditioally coverget, diverget. T F B If a 0 for all N, the a is either absolutely coverget or diverget. T F B If a coverges, the a coverges. T F B If a diverges, the a diverges. T F B If a diverges, the a diverges. O the ext 2, thik of a Theorem from class ad what if b = a. T F B If a coverges ad b coverge, the (a + b ) coverges. T F B If (a + b ) coverges, the a coverges ad b coverge. 2. Circle the behavior of the give series. The abbreviatios are: AC stads for absolutely coverget CC stads for coditioally coverget DVG stad for diverget NOT stads for oe of the others. You ca circle up to aswers for each problem. The scorig is as follows. For a problem with precisely oe aswer marked ad the aswer is correct, poits. All other cases, 0 poits. Series 2 ( ) 2 ( ) ( ) l() =2 ( ) l() =2 e ( ) e Page 3 of 9

4 3. For the followig SEQUENCES: if the limit exists, fid it if the limit does ot exist, the say that it DNE (which is equivalet to sayig it diverges). Put your ANSWER IN the box ad show your WORK BELOW the box. 3a. lim = 3b. lim = 3c. lim = Page 4 of 9

5 TABLE FOR YOUR ANSWERS TO MULTIPLE CHOICE PROBLEMS See Statemet of Multiple Choice Problems for the statemet of the multiple choice. Idicate (by circlig) directly i the table below your solutio to each problem. You may choice up to 2 aswers for each problem. The scorig is as follows. For a problem with precisely oe aswer marked ad the aswer is correct, 5 poits. For a problem with precisely two aswers marked, oe of which is correct, 3 poits. For a problem with othig marked (i.e., left blak) poit. All other cases, 0 poits. Fill i the umber of solutios circled colum. (Worth a total of poit of extra credit.) Your Solutios Do Not Write Below problem 4 4a 4b 4c 4d 4e umber of solutios circled poits a 5b 5c 5d 5e 6 6a 6b 6c 6d 6e 7 7a 7b 7c 7d 7e Extra Credit: Page 5 of 9

6 8. I this problem, you must show your work. Let a = 3! 8a. Fid a expressio for a + a that does NOT have a fractorial sig (that is a! sig) i it. a + a = 8b. Check the correct box ad the idicate your reasoig below. SHOW ALL YOUR WORK. Specifically specify what test(s) you are usig. A correctly checked box without appropriate explaatio will receive o poits. 3! absolutely coverget coditioally coverget diverget Page 6 of 9

7 9. Check the correct box ad the idicate your reasoig below. SHOW ALL YOUR WORK. Specifically specify what test(s) you are usig. A correctly checked box without appropriate explaatio will receive o poits. l 5 absolutely coverget coditioally coverget diverget Page 7 of 9

8 Statemet of Multiple Choice 4 7 ad Scratch Paper Not to be collected. 4. Cosider the followig two series. a. both series coverge absolutely b. both series diverge Series A is Series B is ( ) c. series A coverges coditioally ad series B diverges d. series A diverges ad series B coverges coditioally e. Noe of the others.. 5. Cosider the formal series ad let s N = ( + ) N ( + ). Note that the partial fractios decompostio of is (+) a. s N = ad the series i (5) coverges to. N+ b. s N = + N+ ad the series i (5) coverges to. c. s N = + N ad the series i (5) coverges to. d. s N = N ad the series i (5) coverges to. e. Noe of the others. +. (5) Page 8 of 9

9 6. Cosider the formal series ( ) =2 a. The series coverges by the Root Test. b. The series diverges by the Root Test. c. The Root Test is icoclusive. d. The Root Test caot be applies. e. Noe of the others. 7. The formal series is: a. coverget by the itegral test b. coverget by the ratio test c. diverget by the itegral test d. diverget by the ratio test e. Noe of the others. 7 l Page 9 of 9