mt1_soln - Wl M, Math 210 Midterm 1 Page 1 of 9 Intr: Zeke...

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Unformatted text preview: Wl M, Math 210 Midterm 1 Page 1 of 9 Intr: Zeke Vogler August 17, 2006 Last name:__________ First name: PLEASE READ THIS BEFORE YOU DO ANYTHING ELSE! 1. Make sure that your exam contains 9 pages7 including this one. 2. NO calculators, books, notes, other written material, or help from other students allowed. 3. Write clear and complete solutions including the name of the test you use of convergence/divergence and what your final answer is. 4. Read the statement below and sign your name. I afi‘irm that I neither will give nor receive unauthorized assistance on this examination. All the work that appears on the following pages is entirely my own. Signature: ” You can profit from your mistakes, but that does not mean the more mistakes, the more profit. ” ~ Anonymous GOOD L UCK!!! Math 21C Midterm 1 Page 2 of 9 1. (8 pts) Let 22:1 an be a series where {an},°f:1 denotes the sequence of terms and {snfiil denotes the sequence of partial sums. (a) (2 pts) Write the formula that relates 8n+1, 3n, & an“. m ("(M1) (b) (3 pts) Let an : (2” — 1) x 10. Compute 317 $2, & .93. (c) (3 pts) Let 81 2 5 & sn+1 = 2 * 3n + 5. Compute a1,a2, 85 a3. Math 21C Mi( iterm 1 2. (8 pts) Use a geometric series to convert t ’, 5 i: g, 0‘? ” 734’1004’76704 I :~%[I+mv yearn a hit sen _,§1_g”:4 010 10 l 3. (8 pts) Consider the series 23 (a) (2 pts) Use partial fractions to decompose this series into a series with two terms. 4 .L (WNW) *' (w) Since 'Hnere ’5 ’w ’1 _ _ NJ... _. .__L___, ’7 Mywyj) w} ll‘lffl'Z) 1( 911+?) a g Ve‘“‘0’fi"t‘y fie‘fo‘llowing series ‘ 4 (b) (6 pts) Fin 1: e summmm 1 .i 5n t H ‘— : '1 +1523 ‘— é+$ '\’" 5n : [1,“; 2 v“? "3’9 3.8 into a fraction. 8/ I m; 5’00 / .L L751 if ’0’; + My a) I _.__4___._ 4n — 3)(4n + 5) Page 3 of 9 ?5 w/ diff; KSZ’U va:zM*£ u I'd—'5) ,0 7 w 7 +fl’z5) :7 L/ 3 '3) *erm 0" 2 2 (4n—3 4n+5 75 : 2+%‘+0 = -gztgwa/Mz/Mfi- ll 5' :7 1'5?“ i. 9"" +5 Math 210 Midterm 1 Page 4 of 9 4. (8 pts) (a) (4 pts) State the Comparison Test (NOT the Limit Comparison Test). (ompar/‘ron 7:35} LevL {an be genie)“ an; Hm (ex/2%.; an 5+. ans/9,, an new (same/M & 2—5” convex/g; 3) fan Com/5,76) 1’) I’L H‘m’x CX‘I‘Jk 20/” 5.7L. 4/ San er HEN (swim, #1,.) 46 Zdn dryer/35 ‘3) 25/” XXI/6765. (b) (4 pts) Use the Comparison or Limit Comparison Test to determine Whether the following series converges or diverges 00 n + 7" Z n5/47n n21 f2 603' “ r c [6% l 6"“ Z "5/9 ’mc n97v7" 0%7'1 ’ 71:79 I for a” n {6 £4232, (Ci/Weave; Math 210 Midterm 1 Page 5 of 9 5. (8 pts) (a) (4 pts) State the Alternating Series Test. Alecrna7l/(M/a fer/((95 7795} Far 0 five n a/ILérnwL/rn/ fffft’j 00 Die/Wan : do «a, +6; raj 7» so ,, (0141/8/76; if 7L/nc /&//0w/3y7 )7 (fine/1.52145 Ad//, I) 67,, 70 ifldf' a// VI 2) amgam 1:” an n 2N [Some ml.) 3) ape!) (b) (4 pts) Use the Alternating Series Test to determine Whether the following series converges or diverges 00 Z(_1)n+1%fl ‘LI-H-f-(O if /nn>) 0r n>€ H 3 ,4 :7 15 1 i5 decreas'I‘n/o ‘6’!“ “76’ 5) anHSan 1L" 4/, AZFC] ‘gz' L’H 0” "33 Hm Inn 55 . .1 7 h-voo ’ " I'M —-'4 ‘ V" new , w IMM 00 Hencf/ A/hrnwL/‘n/o 50485 765% Z ("UMP/7," (0)7 V6175}. Math 210 Midterm 1 Page 6 of 9 6. (10 pts) Determine whether the following series converges or diverges 00 n 2 2 n - n ', _, ., Try flaw 75+ or» [EM—2) nl/a) /«Z," 2) , ,v t t’ , I L’H \QQ/LV"/ HM @+/)l/}J:) 3; Km (n+01é’gl Hm 003— ”M ’ hm l 3 "no 73 9'? MM ,1“ :5 n 3: [29% 0:9- '7\ 1 2"”! 1' '3‘ (—2 so Z M0 n (3')) WW?“ I Ly gent/o £29. ‘ V) Hence/ Z (V/)hnl/%) 6671’) Véryé’; by Alfie/rule I ’— v- [Myg/yenw /e 5 7L 7. (10 pts) Determine Whether the following series converges or diverges Z(__1)n< 1717:)71 lgn) M I" m h 93 /n M Y) 77’7/ 200% T115 7" 0V‘ 2:, ,/n(n1) / : 'Z I1 'lfmZ/n _, /I(M -lnn ‘51,:H ZZ I , a - I 4/ new In"? ham “411 545700 ¥.l'// 2’ n¢ 00 50 )n/ [arm/(rigs [007‘ 7,657!“ [VI/ml) 00 "’ In A Flencf/ 2" Inga) Cghwfr/C’) Aida/«Ac (on var/5mg? 725 ,L W ‘ In I" _ [Iv/4| AIL 0r [70:58 inn} _, 1.7” V J— n/ V‘ 0° ) M l c g ‘ :7 Z /'/)[/7"an) 3 17:6'5) WA'CA ’5 fiyéoMC/V‘rc I fer/«6’5 60/ a:/ 1": ‘fi- Hm; Cénvcr765 gfncc lr/:/'—;’;]¢/ ( Math 210 Midterm 1 8. (10 pts) Determine Whether the following series converges or diverges . .. 3 [mm _ /u/V‘ n ._ I h-900 an V haw // "J ‘ 6 \f/ 60 V) 3 ( Hencf/ Z / "/77 fl/WN/flgfi l Page 7 of 9 Math 210 Midterm 1 Page 8 of 9 9. (14 pts) Find the center a, radius of convergence R7 and the interval of convergence for the following power series 3 W18 | S 3 H + 83 3 ( {Mr} "900 ("47)15W’ ’ hm (map 5’ 3/ 5, fl .7: I b/ ear/ICP cam/mafia“ n1 / / w? v.- wc Aavc (0NV6r/agncg wzzm 5’ é] é' )wa/AS— 1:) ~52 x+}d§ 5 Le?" X+3 ’ ig—n 0 '2 % ém/n/b : ‘;,/’). ¢9AV€/}€} /J$c'//r‘5 l Vii/5m I fivlpl) I + X +2: 5' 4/” Led) -) V‘/ a? (:I’Zn Ila/“CA Convg/yCS y :7 I: ‘ may W by )9 “Series ‘5 Cam/52751768 . Ham: / I’hi’érn/a/ 0 7‘ (M (#07614sz /§ Math 210 Midterm 1 Page 9 of 9 10. (16mm) Let, : 8—2; v (a) (8 pts) Derive the .\laclaurin Series (Taylor Series at x : 0) for f(x). I {‘(X) fI/O) Mama/auth Ifer’i‘ejoo/ n ‘70 K... grew) n _?_~_’Xn_2_6i) 0 ('17‘ I 0 [LI X "' o ’1', ~ 0 '/ ‘IX , I ’16 2 —— I o— 2 211 e _ {I} + s .3. +‘/e'2X L/ w x + a! ., -2)! _5/ 3 - Xe (b) (3 39155) State Rum“) (remainder of order n) for f(x). n+) IN) v {(6) I“) (‘3) 6 1C 21%) X 160/ C A’gfl fin (x) 3: @H)" X —~ 71,): (C) (5 pts) Show 111nm.”C Rn(r) = 0 for all x, proving the Maclaurin Series for f(x) converges for all X. 5 3 f X312“)? 7‘7;th 1;”‘e [n [0):0 ’3‘ 'lx —- )( >0 50 0C6 4 Y :7 O > F916 > vQX :Psx'n/ieeif‘nc. > ' H > d I g" (X); 5 ‘ ~———a 0 £5 n am far a. x 0 6M”) \mau {Len/pl A’nac Ag We) ~X<0 50 X<C<0 :, ~1x)~2¢;0 :7 e”">c2‘lc)/ ‘lhcc ex int. ...
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mt1_soln - Wl M, Math 210 Midterm 1 Page 1 of 9 Intr: Zeke...

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