Test 3 Fall 2006 Answers - Math 122 Test 3 1 2 3 4 5...

Unformatted text preview: Math 122 Test 3 November 21, 2006 1 2 3 4 5 Name \<\ ll"; ‘1' 6 Style Points Total g Directions: 1. No books, notes or losing to the Steelers in the last minute. You may CI! use a calculator to do routine arithmetic computations. You may not use your calculator to store notes or formulas. You may not share a calculator with anyone. . You should show your work, and explain how you arrived at your an— swers. A correct answer with no work shown (except on problems which are completely trivial) will receive no credit. If you are not sure whether you have written enough, please ask. . You may not make more than one attempt at a problem. If you make several attempts, you must indicate which one you want counted, or you will be penalized. Numerical experiments do not count as justification. For example, computing the first few terms of a series is not enough to show the terms decrease. Computing the first few partial sums of a series is not enough to show the series converges or diverges. Plugging in numbers is not enough to justify the computation of a limit. You may, of course use numerical experiments to help guide your work, but they do not count as justification for answers. On this test, explanations count. If I can’t follow what you are doing, you will not get much credit. . You may leave as soon as you are finished, but once you leave the exam, you may not make any changes to your exam. The exam has 6 questions and a total of 100 points. Happy Thanksgiving 1. (15 points) (21) Find the limit of the following sequence: { (2:1)”? 11:1 - x , “x ‘ V W N «i {N N W N w gin ma I 3 x ,_ N, WWWWWWWW .3 7 'w- JV“fo was; il ) “I “’1‘?”va ., EX) l H “:3” (if) N “‘5 (7.; “v _ m V i l i [K M VA, A t in M x r in l we WW ,,,,,,,,,,,, x A Oz «N .%M::£:L.:L3{ril :: 51:) s mi N «H 7W”’fi{“‘”“‘"“ _l N «em N we WTTTW ”””” s2 is. W was M N m: x (b) Determine if the following series converges or diverges7 and if it converges, find the sum: 00 J5 n 9 n. .r . ":4. memw, '. w t, “x “v” Z‘j'TQ“ Cg» - rm; 12(“45 <1: mm; ., n20 (A a We .3. r or" fawn“: T”? i” W M Ma if: “ B» l W “l ‘5‘ (3‘ .MmWw M, “A”, " . M... l‘sii (N 4r l l N §< N my l N W l K M \54 z i ii NR ,fi i \V/ «l» x ...- ska J 4; m “QR \ 3‘1"“ V I E '3 3, 2. (20 points) For each of the following series, determine if it converges or diverges. For each test you use7 you must name the test, perform the test, and state the conclusion you reached from that test. 00 1 n . ., 21‘ —-~——~—,—~ DW M9 ewe, ( ) mum/2) «.017 ‘ ‘y ‘ - { a) ‘ {3% If“ EM 42;} m “firearm..- "a" ( W3 AAAAA M j X cwm w r, , f M Te??? {i eat wow >2 ‘ « Qt (b) i (7103 51C} N; ezgggéfig M, f\ ’ . . S ‘ ‘ W M M A "rm: ‘ “d R ééK/VY‘} i Q j}: {‘3 M59 (J: N :K k X :33ng C 6‘“) “1” J} "T5351 we, cm (W W)? (k m g 00 2 s‘n2n w r. K n“??? 3?: ((1) Z Cami: wAii mm.) r“ k a K #3:) q \ [I N " h""’vrww _ r. .\ 2.1%» “is \N N g “fry-W ems w a < t“ Fm ‘ C53 m g) m: t .2 CM “MWMW “* ~' « t "322’ g” M N? "a: t ,3 3. (20 points) Determine if the following series converge absolutely: con— verge conditionally, or diverge. For each test you use, you must name the test, perform the test, and state the conclusion you reached from that test. 00 2 F” n + 4n + o 3; gig) wax»; d) ~__1 n ‘ ‘u M N m M1 ()g( )3n2+3n+2 \ more r , v _ liwfif k “gag—yaw“ ‘fljmmjj_:miwm “3‘— :E: Q (s3 l ‘3' "a ‘ ~ ‘ ~ 2. _ ‘ 5;», a ““ Kris“ 4: all 00 (_2)n “ E t W; M, 1 CE. R Q 6:“ (b) 2 n, (draw \reléi.>t Mi? l M; J a 71:2 ' W Marl; K J? a o \ WM/ “nix ” a , \a it???“ N Asa Ce» U“ “H 1"? 654:: “I ‘ m “f”‘””"“"‘““ A ‘ g" m ML, all >1 ~: gel, rwa l 33” V a 7 J a ~~~~~~ ~~ 53¢ lwferfimm W a» “it liar“ >5 v g ‘5 WWW "' WW“ 3 V W WWWWW : "K k MW «mmmw w» w A M lat 3. " lwr @733“ {J “1/ \ r“7“”*M§”3 M g4 “tar: Ma ““ H») g Winn-l, 00 I 8271 (A {D ,r d ~1 7“ “5(ng \1 “2:; < > ( > gamma: wwwwwwwww 7 p f" \ E g) HEAR) f") W W WW WQMWX ” 1‘1“ “ W N Q/Wn w 4. (15 points) Consider the power series: (3 — f6)” n + 1 n22 Where is the power series centered? '25 W C (b) Find the radius of convergence. N‘t} (rang) w») A \gmbd <1 (I '2, QM W M "’r .3} N “5% CC} (C) Find the interval of convergence. [Hintz Check the endpoints.] f, '~ ' " x ML” bx v. L.“ CT W 6:“ xix“: \ N :7 El 5 N CK! * é LN- C‘QN‘J CQND a N. me i N ‘3: 31 SE 5. (15 points) For f (93) = 0080112) (a) W'rite out the first 4 non—zero terms of the Madaurin series for fix)- 4 l4 5; ELK L“ (GA Li 9 hi Cog) (' :1“ i “— 3:” “t mg...“ W W " “ " 2% I4 2_ (a) (1)) Find P4 (an), the fourth degree Madaurin polynomial for f w - m. A» we \ a 4 ,2 :1: cos :1: — 1 + (C) Compute 11m —(——>—8——~—~_2: m——>() 33 You must ShOW your work in order to receive credit. 3 I X , a a; e /“ gwawi ~ m/e/{f ELM A “ESQ; (o L / i 6. (10 points) Indicate whether the following statements are true or false by circling the appropriate letter. A statement which is sometimes true and sometimes false should be marked false. 00 If 0 < an < b” and Z an converges7 T e3 then 2 bn diverges. n:1 If the nth partial sum of the series 2 an is @ F 00 Sn : yL—Z—Z, then 2 an 2 1 7L:1 00 CO If the series 2 an diverges and the series Z bu diverges, 71:1 'nzl 00 then the series Em” + bn) diverges. 11:1 T® WE 00 If lim an 2 0 then E an converges. ’(‘lf—FOO 1 n: 00 If Z on converges then an 2 0. ® F n:1 ...
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