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exm4soln

# exm4soln - Fourth mid-term examination Math 32 Fall 04 2...

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Unformatted text preview: Fourth mid-term examination / Math 32 / Fall 04 2 Problem 1. [15 POINTS] Determine the radius and interval of convergence of the series 00 215—4" Z( )0 ”:1 n5” ME AW’WAQ “(ho AH on; Am“ (:3: jg )21‘“) a IZDL'Q) Rim / x (5),) m“! “V, I,” '5‘ 1‘ M new GM Ina” “if; 7 I‘m” 5 NH 5 825:9“ n6“ } _ 45 memo/9 converge“ 12/723141 e W“ 5/7. 5/2— Wﬂég «.3, €9.ch 4; WW lath/L :;:W:::: .. L a/L NOW A 47% (Mel poi/WE 6f 7C 1 4/; n ,_ f\ ' . ()1: i 9.7%“: 9., 2:9. Amy/mg Apr/weak ”gel/W). an N v go CQAUE + \7‘ No 9w ’ L wwnig [ﬁe/{m 7L ’ L L £7,144 3" J. dwarf/ﬂ “w" ) 1 4’7? ’/ A I Ab Radius of Convergence Interval of Convergence W 7.. L‘” -v..-AUJ , / , 2/ A lyl (5pts) Give the general formula for the Taylor series of the function f(\$) centered at a. (’0 w Cm A 1 4 a) (\$4) 4“” A“ Ti 2. (5pts) Find T3(:c) (the third degree Taylor polynomial) for f(:r) : eI centered at a : 2. 2. 3 / ”a (ac—a) + (2m) (2%) l}(j‘) e ( l l 7/{4 + 3\ > 3. Km Tavtob wow (4) 2. gym 5 f OJCJL—af : S7001; m, l .— " C: L 5 ~ ‘ WK 70§\$IlylL QYVD/ Wl)l 51 (”MA 2, WZ'D é 9 2-5 LT I+\ Fourth mid—term examination / Math 32 / Fall 04 4 Problem 3. [15 POINTS] Derive the Maclaurin series tan‘1 (2:) using the Maclaurin series formula for other known series. Compute the radius of convergence of the resulting power series. Show all of your work. /- l u”: :1 g“ ”‘4‘ 1+9 ‘ A/O 9L )C PG Ln+\ A <1 {Elli} :- .'m At : Z—i it. QCV ’6 0 1H? pm 0 O n90 m 73H 4) l ’4' Coax/Cr Cc 2l : bfl Y ”helm”? 00’ 5m OI Fourth midwterm examination / Math 32 / Fall 04 Problem 4. [20 POINTS] Solve the initial value problem 3;” + .93; = U, -y((]) z i}, y'm) : 3 usii'ig power series. Show all of your work. "‘ ﬂ [0) : Co :0 Mi 2; Ch?» ‘3 a g; A, ; a: 2 3 Ufa” fngx ‘ 3(0) ' r; :1" n.—| : CMZ) (n+0 CM?!)L [P‘- 2 [ ~ ) Ch I 2 K3 ) Zn M m 4 A rmwsswn YaleO/N £»b\$%Mrj In ﬁt ohﬁgrcﬂ7gﬁl QT“ wc‘ 2L mlﬁmLi . L D > C :D grail/1 q. “:9 _ 7‘ 2n 9&0; C050 g C;:0 CE 0 3 ,— _.. ... "Ci 3 : Gill—:3: 3 1 q 4 60;; l tofu. C5 1 :51: ‘53 F 6 tr 3‘ 5'5” 7 a Mr} E — _ r 3 2 4 FF. (.7 _, £2} £5 _, L‘ 3.1.. F Can-H C)@H)l 39- ZM"! m 0J0! ’16 {W Suww UCL) bi" 1524\th L by Fourth mid-term examination / Math 32 / Fall 04 ('5 Problem 5. [20 POINTS] Determine whether the following series are absrﬂulely com-er— gemg (“.011(1il.-i(')Il‘rlll}-’ convergent. 01‘ divergent. Justify your answers and give ﬁg: na.me[s) of f_'ﬁ4l1~L€)<G/V\- 'Ra’ho 402+ Em” Vii?) a 2:», r53)?“ \$0 F CEO/wager“?L A’Do-b a“ A Oh 4. (5pm) 30—41111 1(n) 12131“) f A 03.75 uq L "Hr—2“ Z A s) P gam'w 'FL—-] _.—. __L_ LIMI+ {DW‘Sw/‘x 48/391 . 601’“th 90% 2,- n1 “ d—ﬁ J.9 [VAC-i (GIN 5‘” ,Prm {ii/CE) 9W") JE‘ICA) "-1 “2— a '50 ‘9 ham 7;“ :, Nam nl- :. Avon ConVegcx-A. Fourth mid-term examination / Math 32 / Fall 04 8 Problem 6. [15 POINTS] 1 n 1. (4pts) Consider lhe series 2:: .0 10,1 . How many terms are needed to compute the sum of the series with an error less than or equal to 10—6? What is the exact value of the summation? B H I: , ‘ —-1 _ 0 n4; 5N0) (Tie #1 jé-len i5 gmﬁb‘eﬂ'} ’11? 9'“; M at 10" ﬂ-e ciQ/DlYed atﬁuvccb é farm ﬂ {ZMC‘i game/mica 913, 2:6? ﬂif 1 _ L9 A90 1" 1%,) .— H // 2. {4pts) jelnpute the summation i0:[ ) win—H ____7-J\ _ 71 "F“ I 2 F ,_ F1 r’" n__0 (n) F )1 ZLJ‘ Q”)! 5 T 605.? :3 #17 \$55" 3. (4pts) Consider the sequence {Q11} where Q.” = 23:40? Does this sequence {Q11Q2,Q3?. . .} converge or diverge? If it is convergent, ﬁnd its limit. Ties. show is 4&- pav-ﬁal 5mm same (A ’1’; game) ”2;; leri {)7} Jim gdfmflfi. 13 H7234 _‘_ e2— “ o! ' .f'o U u 4 (3pts) Given 2 an— — .10. compute limn_ .90 C glefz i-QA 1% Convcggn+/O 1‘7l AMANL fringe/J fun nape 47071 Q4 :— m e" _ ‘ ﬂ’ﬁos u (b n \ ...
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