Test 3 Fall 2007 Answers

Test 3 Fall 2007 Answers - Math 122 Test 3 81 l 2 i 3 4 5...

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Unformatted text preview: Math 122 Test 3 81: November 20, 2007 l 2 i 3 4 5 Name m l<E \r 6 '7 L t *l , P ‘ S y e omts Total Directions: 1. No books, notes or dressing up like a turkey. You may use a calculator U! to do routine arithmetic computations. You may not use your calcu— later 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. Happy Thanksgiving 1. (15 points) (:1) Compute the limit of the following sequences: 1 {(J J71 F371} I 2 +17 ‘n:1 N N N"*’.-—_ Mill?) fiWLW) L1>N2~O N ,N ,9 New: a TD” N 0‘3 Q7692“; +CY;)N n21 S‘NZ(JN~) ' NZSINZ C-L 1“ vQLvm “m QJW‘“ W) New "‘63“- N’aw (20$ L> 2 :— % (g m fill) «£1 gilt“:” Nata) Nae: a"; (b) Determine if the following series converges or diverges, and if it convergea find the sum: 214 2- (2H,) 0.0 0° to l §g5=éei:7::“‘ N21 N“ 2“ 2. (20 points) For each of the following series7 determine if it converges or diverges‘ For each test you use, you must name the test, perform the test, and state the conclusion you reached from that test. 0° n+1 2 ”WWW”— l r, { oNV‘. (d) Z<n2+2> C 71:1 (p.144 “32‘ LL?!" Jaw-W Em :— :L (Cl) i0: \/ arctan n D \ \{l 7121 N4“ ,ng‘ REWN“ :- 3" fagm 53-90") 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 test7 and state the conclusion you reached from that test. DC n2 +471+ 17 t «1 ”w.—-——————-—- \1 . R85 (8) 2:); l (.712 +3” +5)? (ION fin“ Matthias H"! Wm .. ——_.——.._.____ 77:2 hm ‘ > 1 DW 60 “To MT Com? TE‘ST JLMN N ‘1 l x/ M Ml :30 \/ "Hm—W < M RSV" New van!“ UQMHHS Mn 00 13! (C) Z(_l)n+igfi D \V 77.22 N —9C® 00 n! 2 (d) 2;" VIOLA)? Comv RES 52E 1C 4. (15 points) Consider the power series: £121.23”..an 2-4-6-~(2n) 71:1 (a) Where is the power series centered? C2263 (b) Find the radius of convergence. is. WWMW we. N "’1’ 0° /M;fifi(aw +9.) mwrwtaf) x/W as; (c) Find the interval of convor.gence [Hintz Check the endpoints] 00 m x . mm?) MR N a“ =2 l Jim W 1.11.” (3W3 ? \>0Q MOI N=§ N3 GO 7&0 / ZERO ~M QQ _ __ w / . . (‘3‘? = fc~ur~x D‘ V m X E 3.»;1»{ae) :) ( #1 )g) (d) True or False: If 2 0,7150” converges at :13 = 2 then 71:1 i. it must converge at m : —1 6‘) F ii. it must converges at a? = 3 T @ iii. gigglc an =0 G) F 5. (8 points) For f<f13> = v.7: + 1 (a) Find 173(1") the third degree Taylor polynomial at a : 0. RV): (rm-it)”i L \ l 3 a -1 _~ 2 __.. F'ur—iCm—D Jr; PBCKW=1+év<1~%X 1",??? x _ ’15 -—..L ‘ F—"c x3 :5 *«L ( w) «x - 2 H ”‘5' 1:. \ + 11. "'" .X + 34—? W 3 2 2 l— —- P we gum e 8 lé (b) Find R3(r1:) the remainder for the third degree Tayicn' polyrrmrrial atazO. .57 ‘1 ~55 ”2‘.” 12350 -~—- £132, (bow m: TCCgm w W. q, 6. (7 points) (21,) Find the Maelaurin series for sin(a:3) 5 5 7 SiNCXcB‘J— 74"”..éndrx “X.~“' '5‘. st “”11, °\ \5 21 3 __ ”'3__ X ‘k/ __)( ’- 5‘N(K> "“ 7< 75;? + 5;. 757 _ , sink?) ~— 3:3 + (909/6) (b) Find £1115 ——-————;—1—O————- C 7. (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 3 an < b.” and 2 an diverges, then a) ":1 T @ 00 2 bn converges. 71:1 00 If an > 0, bn > 0, and 2 an diverges, and 11m ~17— : 0, n2] nfloo bn b) m . (,9 F then Z bn diverges. 7121 C) sequence {[371}le diverges, then the sequence {an + 571K; diverges If the sequence {an}:1 converges and the @ F d) If 2 an diverges then ”113010 (Ln : 0 T ® 7L:1 sequence {J} converges to 0. If the sequence {on} is bounded then the ® F n, ...
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