Fall 2018 Exam 2 PIN: 17 INSTRUCTIONS

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1 MARK BOX problem poits 0 0 HAND IN PART NAME: Solutios PIN: 6-3x % 00 INSTRUCTIONS This exam comes i two parts. () HAND IN PART. Had i oly this part. () STATEMENT OF MULTIPLE CHOICE PROBLEMS. Do ot had i this part. You ca take this part home to lear from ad to check your aswers oce the solutios are posted. 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. Durig this exam, do ot leave your seat uless you have permissio. If you have a questio, raise your had. Whe you fiish: tur your exam over, put your pecil dow ad raise your had. Durig the exam, the use of uauthorized materials is prohibited. Uauthorized materials iclude: books, persoal otes, electroic devices, ad ay device with which you ca coect to the iteret. Uauthorized materials (icludig cell phoes, as well as your watch) 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 durig the exam (it will be retured whe you leave the exam). This meas o electroic devices (such as cell phoes) allowed i your pockets. Cheatig is grouds for a F i the course. At a studet s request, I will project my watch upo the projector scree. Upo request, you will be give as much (blak) scratch paper as you eed. The mark box above idicates the problems alog with their poits. Check that your copy of the exam has all of the problems. 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 Math 4

2 0. Fill-i the boxes. All series are uderstood to be, uless otherwise idicated. 0.. For a formal series a, where each a R, cosider the correspodig sequece {s } of partial sums, so s k a k. By defiitio, the formal series a coverges if ad oly if the lim s coverges (to a fiite umber). [also ok: the lim s exists (i R)] 0.. Geometric Series. Fill i the boxes with the proper rage of r R. The series r coverges if ad oly if r satisfies r < p-series. Fill i the boxes with the proper rage of p R. The series p coverges if ad oly if p > State the Direct Compariso Test for a positive-termed series a. If 0 a c (oly a c is also ok b/c give a 0) whe ad c coverges, the a coverges. If 0 d a (eed 0 d part here) Hit: sig the sog to yourself. whe ad d diverges, the a diverges. 0.. State the Limit Compariso Test for a positive-termed series a. Let b > 0 ad L lim a b. If 0 < L <, the [ b coverges a coverges ]. If L 0, the [ b coverges a coverges ]. If L, the [ b diverges a diverges ]. Goal: cleverly pick positive b s so that you kow what b does (coverges or diverges) ad the sequece { a b } coverges State the Ratio ad Root Tests for arbitrary-termed series a with < a <. Let ρ lim a + a or ρ lim a. If ρ >, the a diverges. If ρ <, the a coverges absolutely. Prof. Girardi Page of Math 4

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. The symbol is uderstood to mea. Scorig this problem: A problem with precisely oe aswer marked ad the aswer is correct, poit. All other cases, 0 poits. O the ext 3: thik of the th -term test for divergece ad what if a.. T If a coverges, the lim a 0.. T If lim a 0, the a diverges..3 F If lim a 0, the a coverges. O the ext : thik of def. of AC/CC/Diverget, the Big Theorem AC covergece, ad ().4 T A series a is precisely oe of the followig: absolutely coverget, coditioally coverget, diverget. is CC.. T If a 0 for all N, the a is either absolutely coverget or diverget..6 T If a coverges, the a coverges.. T If a diverges, the a diverges..8 F If a diverges, the a diverges. O the ext oe: thik of a Theorem from class ad what if b a..9 F If (a + b ) coverges, the a coverges ad b coverge. O the ext : thik of a Theorem from class ad what if f(x) si(πx)..0 T. F If a fuctio f : [0, ) R satisfies that lim x f(x) L ad {a } is a sequece satisfyig that f() a for each atural umber, the lim a L. If a sequece {a } satisfies that lim a L ad f : [0, ) R is a fuctio satisfyig that f() a for each atural umber, the lim x f(x) L. The ext 4 are from a MML homework problem.. T The a coverges if ad oly if a coverges..3 F If a coverges, the (a + 0.0) coverges..4 F If r diverges, the (r + 0.0) diverges, for ay fixed real umber r.. F If coverges, the coverges, for ay fixed real umber p. p p0.0 Prof. Girardi Page 3 of Math 4

4 . Geometric Series. (O this page, you should do basic algebra but you do NOT have to do ay grade-school arithmetic (eg, you ca leave ( 8 )9 as just that.) Let, for N, N ( ) s N 3... Below the box, usig the method from class (rather tha some formula), fid a expressio for s N without usig some dots... or a summatio sig. The put your aswer i the box. ( ) [ ( ) ( ) ] N+ s N (3) (aswers will vary) Fid a expressio for s N rs N, which results i a cacellatio heave. substract ( ( )) s N ad so [ ( 3 ( A ) ) ] N+ s N ( ) [ ( ) ( ) 8 ( ) N ( ) N ] s N ( ) [ ( ) 8 ( ) 9 ( ) s N 3 + N ( ) N+ ] ( ) [ ( ) A sn s N 3 A [ ( 3 ( ) ) ] N+ A ( ) (3).. If 3 ( umber it coverges to i the box. Justify your aswer below the box. 3 ( ) [ ( ) ( ) N+ ] ( ) ] N+ ) diverges, the write diverges i the box. If 3 ( ) coverges, write the ( ) (3) ( ) The series is a geometric series with ratio r ; hece, sice r <, we kow the series coverges. The series will coverge to lim N s N. Sice 0 ad so <, the lim ( ) lim s lim (3) N N [ ( ) ( ) ] N+ ( ) (3) [ ( ) 0] Prof. Girardi Page 4 of Math 4

5 3. Below the choice-boxes (AC/CC/Divg), carefully justify the give series s behavior. Be sure to specify which test(s) you are usig ad clearly explai your logic. The check the correct choice-box. X absolutely coverget coditioally coverget (caot be sice it s a positive termed series) diverget Thikig Lad. big ad (p-series, p > so) coverges. So we try a compariso test. Note for each N so if use DCT, comparig the give series with the coverget p-series, would be boudedig below by a coverget, NO GO. So let s try LCT, comparig the give series with. Ed of Thikig Lad. Let The a a lim lim b ad b. lim lim lim 0. By the LCT (Limit Compariso Test) sice 0 < <, the series ad do the same thig. The series (it s a p-series with p > ) coverges. So the also coverges. Prof. Girardi Page of Math 4

6 4. For a atural umber >, let a ( )! (3)! 4.. Below the box, fid a expressio for a + a i it. The put your aswer i the box. that does NOT have a fractorial sig (that is a! sig) a + a (3 + ) (3 + ) (3 + 3) a + a (( + ) )! (3 ( + ))! ( )! ( )! (3)!! (3)! ( )! ( )! (3 + 3)! (3)! (3)! (3 + ) (3 + ) (3 + 3) (3 + ) (3 + ) (3 + 3) 4.. Below the choice-boxes (AC/CC/Divg), carefully justify the give series s behavior. Be sure to specify which test(s) you are usig ad clearly explai your logic. The check the correct choice-box.you may use previous parts of this problem. ( )! (3)! X absolutely coverget coditioally coverget (caot be sice it s a positive termed series) diverget ρ lim a + a A lim a + a part previous from above part lim (3 + ) (3 + ) (3 + 3) (3 + ) (3 + ) (3 + 3) A (3+) 3 (3+) (3+3) A 0 ( ) ( ) ( ) (3) (3) (3) 0. Sice ρ lim a + a 0 <, by the ratio test, the series a is absolutely coverget. Prof. Girardi Page 6 of Math 4

7 . Below the choice-boxes (AC/CC/Divg), carefully justify the give series s behavior. Be sure to specify which test(s) you are usig ad clearly explai your logic. The check the correct choice-box. () + X absolutely coverget coditioally coverget diverget This was is umber 4 from the Serious Series Problems. Over for aother sample studet solutio. Prof. Girardi Page of Math 4

8 .sol. Aother sample studet solutio. Prof. Girardi Page 8 of Math 4

9 MULTIPLE CHOICE PROBLEMS Idicate (by circlig) directly i the table below your solutio to the multiple choice problems. You may choice up to aswers for each multiple choice problem. The scorig is as follows. For a problem with precisely oe aswer marked ad the aswer is correct, poits. For a problem with precisely two aswers marked, oe of which is correct, poits. All other cases, 0 poits. Fill i the umber of solutios circled colum. Table for Your Muliple Choice Solutios Do Not Write Below problem 6 6a 6b 6c 6d 6e umber of solutios circled B x a b c d e 8 8a 8b 8c 8d 8e 9 9a 9b 9c 9d 9e 0 0a 0b 0c 0d 0e a b c d e a b c d e 0 0 Total: Prof. Girardi Page 9 of Math 4

10 STATEMENT OF MULTIPLE CHOICE PROBLEMS These sheets of paper are ot collected. Abbreviatios used: DCT is Direct Compariso Test. LCT is Limit Compariso Test. AST is Alteratig Series Test. 6. Limit of a sequece. Evaluate 6sol. lim Cosider the formal series + 3. sol. This is the first problem from the 38 Serious Series Problems. Note that +... ufortuately, some folks told me they were equal Cosider the formal series () ( + )( + ). 8sol. (+)(+) big ()(). So let b ad a () (+)(+). The a lim b lim ( + )( + ) lim ( + )( + ) lim ( + ) ( + ) lim ( + ) ( ) + ( + 0) ( + 0). a Sice 0 < lim b <, by the LCT, b ad a do the same thig. We kow b (p-series, p > so) coverges. So a coverges. So a is absolutely coverget. Prof. Girardi Page 0 of Math 4

11 9. By usig the Limit Compariso Test, oe ca show that the formal series. (9) ( + ) ( + ) ( + 3) ( + 4) is: 9sol. Let For sufficietly big, a So we let b a b a ( + ) ( + ) ( + 3) ( + 4) ad compute ( + ) ( + ) ( + 3) ( + 4) [ 4 ( + ) ( + ) ( + 3) ( + 4) [ () () () () ] /. ( + ) ( + ) ( + 3) ( + 4) whe is big () () () () 4/. [ ( + ) ( + ) ( + 3) ( + 4)] / ] / [ ] / ( + ) ( + ) ( + 3) ( + 4) a Sice 0 < lim b <, the LCT says the a ad b do the same thig. Sice b is a p-series with p, the b diverges. So the a diverges. 0. Fid all real umbers r satisfyig that r 6. 0sol. Sol: ad 3 First ote that for the series r to coverge (so that the problem eve makes sese), we eed r <. So let r <. Next, to fid the sum r, cosider the partial sums s def r + r r + r. Cacellatio Heave occurs with a geometric series whe oe computes s rs. Let s see why. s r + r r + r r s r 3 + r r + r + Do you see the cacellatio that would occur if we take s rs? substract s r + r r... + r r s r 3 + r r + r + ( r) s A s rs r r + Prof. Girardi Page of Math 4

12 ad sice r, the s r r + r So we are lookig for r R so that r < ad [6r + r 0]. The series () + l, sol. is r ± + 4 (6) (6) sice r < r r r. [ r. Note r r 6 r 6 ± { ] [6r r]. What is the smallest iteger N such that the Alteratig Series Estimate/Remaider Theorem guaretees that () N () 0.0? Note that sol. Note that 0 So 0. () 0 so the AST applies ad tells us that () N () (N + ). () Note [ N () (N + ) 0 If N 3, the (N + ) (3 + ) 4 6 < 0. If N 4, the (N + ) (4 + ) 0. have (N + ) wat 0. ] [ 0 (N + ) ]. coverges ad that Prof. Girardi Page of Math 4