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Sol-166E3-S2008 - MA 166 Exam 3 Spring 2008 Page 1/4...

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Unformatted text preview: MA 166 Exam 3 Spring 2008 Page 1/4 10—DIGIT PUID RECITATION INSTRUCTOR RECITATION TIME DIRECTIONS 1. Write your name, 10—digit PUID, recitation instructor’s name and recitation time in the space provided above. Also write your name at the top of pages 2, 3, and 4. 2. The test has four (4) pages, including this one. 3. Write your answers in the boxes provided. 4. You must show sufficient work to justify all answers unless otherwise stated in the problem. Correct answers with inconsistent work may not be given credit. 5. Credit for each problem is given in parentheses in the left hand margin. 6. No books, notes, calculators, or any electronic devices may be used on this test. (12) 1. Determine whether the following statements are true or false for any series 2 an and 11:1 00 419% ”CL 2: bn. (Circle T or F. You do not need to show work). NFC "=1 00 00 (a) If 0 < an < 1),, for all n and 2 bn converges, then 2: an converges. I? J ‘ (I) F n=1 n=1 wmpmisofi e . 0° . (b) If "1:120 an 2 0, then ”21 an converges. a}! -% oU°M°§$ T ® (C) Hoe an is convergent, then —1 "an is conver ent. F 1 1 g n: n: % euro... U3 ab$obMfl mnulflymt 0W“: ”mm some???“ [El my _ ., .7 (12) 2. Determine whether each of the following series is convergent or divergent. (You do not need to show work). 00 4 pi}; each 0" L L a) : <—+sn-H> = 2 "22, 27. +32} w Y) n Y) ‘L mu 65% [1‘73 0.. r = «3% ,n =2 1.1 > A n DO (b) X: n Com me with Z 3— “ luck vw. n: (n+1)2 P hi! (2.) w (:2 COYlUQ-T m1: W 1: i eom 513K 2 % (““32“ g) {OvaUJ7>l IL] < 1 ‘3) ..) 2.. n __. (C) -— =—3-—-——+Z "1’7 . 712:; n3 + 277,2 _ 1 10:3 n3 «Rm —1 diueVGg-n‘t " (LL’U. by Hm. .temb 1:681- " CDMPOWQ‘, W‘d‘ Zia- ns'b H MA 166 Exam 3 Spring 2008 Name _._—— Page 2/4 (30) 3. Determine whether each series is convergent or divergent. You must verify that the box. °° 1 (a) 2 11:2 71 Inn I \‘l ‘ *9 check war WW . W‘memaa ’ uw_ Show all necessary work here: inteoé‘vqa te. 511 j 1 49‘ 2. 1 12901 (x =_.._—-—- la mht\nw0ut$ OSktfilfi «MA otuvvcs 2 00 g ) ‘3! arm ,l’J moémf)) co @ j—i—CQ—DUL’X z/Qim St 1‘ <10: 2km [EM]: 2 'x 21m t—>oo 2.me t—am @ @ :L'm EZWMt - Nana] : °° t—ww Cntegm‘i U> (Madge/ht 0M .'. Seuss m «Liwevnabwvi l: ' Tabéérm W MA 166 Exam 3 Spring 2008 Name _—____ Page 3/4 Show all necessary work here: A [tgx’nmtlwékz S ewes test By the alierwntwfi Veem'e3 test, the series is (Dnuex (gem: (12) 5. For each function f , find the Maclaurin series and its radius of convergence. You may use known series to get your answer. (a) f($)=$62"’ 00 n e": Z .71.. ~oa<x<<>a mo n' 9'7 0° ‘o n e :. X E 2___)—%<fi<4” n20 “‘ 0° 1. 2X . Y] “Ii? < X400 xe : Z 2 7} , ~°° w=0 n (b) f(a:)— — sin(x2) 2n+L Sinx: j::§£l_21___ V" O (‘Rm 1’1)! )«cm x «c MA 166 Exam 3 Spring 2008 Name — Page 4/4 00 (16) 6. For the power series 2 n3 (a: — 5)", find the following, showing all work. n: 0 (a) The radius of convergence R. Y” 1 ® R61") test: an+1:(‘fl+i> (st—S {: (L+1)3IX 5|_.> ”~51 )cvih—mo OLA "3 (X- 5)“ wies convexgw I» I’X——‘5I<i 0" 4<X <G @ 0) M diver?» \f l><—‘>'|> 1. (b) The interval of convergence. (Don’t forget to check the end points). 2100‘” comer?” { 4 < x 4 ‘6 ® km Z” E " oil in Jew, a :ajm (-0"? PM? W 79:43 “‘0 h (:1) ueche/x mun W9” n 71—909 Wham 7(26 '. i n GUM?” Interval of convergence (N: 53“? 9) 7. Write out all the terms nof the Taylor series for f (1:) _. 1 + :1: + 51:2 centered at a = 2. £09: gig—l filflx-9~)= i—(fl—a— d§(2)(x .2) +—_. Q” (2) (103232 +'-‘ Rx) -. l+x+x H02.) :17 96¢): 1+9»: £22.) 2 6 2. £15) '2. (2:) :‘2 7 -\— 5 (x -‘2) + (xwz) {who =0 . i'mh) :0 (5) 8. The Taylor series for f (cc) 2 lnz: centered at a : 2 is n— 1 2”1(n) f(:1:)= lncc— — ln2+ Z( (1: — 2)" Find f (166)(2) Leave your answer in terms of powers and factorials. in) ca) __ (—W’i —— (3166; Zilacén; 1-, NFC 166‘) ' : ZLGC (166)- - 2.16::{150 f(166)(2) : __ 4.6 5 i . 2165 ...
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