Midterm 2 - Hiday, March 12, 2010 Calculus 1501B Page 1...

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Unformatted text preview: Hiday, March 12, 2010 Calculus 1501B Page 1 Second Midterm Examination 1. Consider the sequence given recursively by 1 2 a1=2, an+1=-2- an+-—— . 2 (a) Calculate a2 and a3. marks NI- do u N\v' a1: Ji(q‘+% a. A; (2+;V' “5 ‘ Had %;\= ‘50? 3'92.) ‘3 éC-ZH 4A) “#430 g \‘7 'fi. 6 (b) Assume that {an} converges, and that lim an = L. Find L. mar/cs 7H°° imam =L. win- 50 kmiOM‘ :- I. “an. 1AM 6m“ = 1W 1.; (cm «'1: “9.. MIN. a“ tthmA“ + j;— 1 W". 1mm.» h-lvo. Calculus 15013 Friday, March 12, 2010 Second Midterm Examination Page 2 8 2. Determine whether the sequence given by mar/cs [ln(n)]2 n an: converges or diverges. If it converges, find its limit. 1”“ an: LHM LJ?’ “'9” “‘9” n .. o». )‘ .2: - 323:” L12. L9,] ; lMM 2&%?(“;) x3.» -.- 1w L912 xav \c ; 2AM 9. X*°‘ \ '3 km 3; xa» X =0 law «We: +° °' Hiday, March 12, 2010 Calculus 15013 Page 3 Second Midterm Examination 4 3. (a) State the e — N definition of lim an = L. marks ""°° FM MW 579’ Than. n a. mwwaGN 5O d1 “>N’ah‘M \A“_L\‘$ 4 k (b) Prove, using the definition asked for in part (a), that $133055; .= 0 ?ru€ R, k wrrk "T, .t e >9 r “‘5 ‘/ 4- -o\ ‘ i choose. N = E e (S \ M“ t N 3:“ ‘ i h.) . L" ‘/t M“ > t Tkw h. 5 e y Vt. v.) e. ’- a .1. TM AM!- i so fih-ol‘i Calculus 1501B fiiday, March 12, 2010 Second Midterm Examination Page 4 8 4. Determine whether the series marks 00 (n + 1) E 1n n eQMz-Moa- (“3’9M'7’3b' ' Friday, March 12, 2010 Calculus 1501B Page 5 Second Midterm Examination 2 5. (a) State the Monotone Sequence Theorem. marks Evevs boundcll mano‘l‘am'c. Sezuewca \$ Cave-(used. 8 (b) Suppose an > 0 for n = 1, 2, . . . , and that marks " 1 sn=Zalc < 2—; 16:1 00 for n > 1. Prove that 2 an converges. n=1 ? PM; an» 3w n=n,2.. —- Snag sh+qn>5n gut n=“1,-. Tull-s ‘5 “mcw’ééwg M50 $.12-l=l “5"42 5° 15 an. Fm “4,1,... TW‘ {$55 ts hounded. Hence {Sm} taxsz 53 m Mamba". $¢Bmu Tkuvem, I; .’ ion“ tam-£44,, Calculus 1501B Friday, March 12, 2010 Second Midterm Examination Page 6 8 6. Determine whether the series marks {If n3+n2—3 n=1\/n9+7n3+3n converges or diverges. Justify your answer using an appropriate test. _ s 1_ a Let am-Lib__3_ M. bng =13 Mums" Jm “m .3 J. n"! 1"" 2.1.; 1‘,“ “inf-3 “a. h “'9 “ " \) Maw-r3“ V“ 5/1. = Rm n,“ (uh Ive-’3) h. 4' 704-3“ 9 'film 7‘ h (“3+ ni-S )6;/: “An JWQ+7M+Sh /Y\"II. =' 1m c “an |+ 4' 5/53 7 2 |+0-O 8 marks Friday, March 12, 2010 Page 7 Calculus 1501B Second Midterm Examination 7. Determine whether the series 00 212 “=2 n (1n n) converges or diverges. Justify your answer using an appropriate test. .L. _L__ Minn)" M 30“: xMMx)?’ Let Qn= C m 904m“, whmm, mam“ Mttfifi) (“HAM = 9AM S“- "L—‘h" a. t-‘ooo 1 thx)2 {mm we u: 9w» w" 3‘?“ flaw .\ = 97+; s ‘4. -_,_L “ —J. .L 0(1X(&~XJ"M' m1 LAWN- Thaw. 9.» ‘J—M=,Q -.L s, ' if": [1 “‘9‘ +an he A”. J. =°+hz Calculus 1501B fiiday, March 12, 2010 Second Midterm Examination Page 8 8 8. Determine whether the series marks °° n + 4H 23“ 71:1 converges or diverges. Justify your answer using an appropriate test. n b“=§_;=<%)h dwevg Tesi’ 0k v. -" +4" _. n. 4:. 9.....- - a” a) J 4'“ op. Lox annfi n+3 Tm “Atl\- “it” qua-4"“ M gm“, ’ saw) 3“ at run 1"” __,o+—4 =7 “3:340 Mom 3 Friday, March 12, 2010 Calculus 1501B Page 9 Second Midterm Examination 8 9. Determine whether the series marks i n4” ' 77. n=1 converges or diverges. Justify your answer using an appropriate test. Li 9‘; Am a Q =L%§_)n = WNW“ N h“ (mm-1.3 (“-5) (“43 .. .. a = V‘ n n ‘ M rw‘ n-1. n-5 &-4)(n.¢)...g Calculus 1501B Friday, March 12, 2010 Second Midterm Examination Page 10 8 10. Approximate the sum of the (convergent) series marks 00 : <—1>"—‘ L, n=1 (2n). to within an error of 0.01. Leave your answer as the sum of fractions. M n... Let 5 = 7. C") J... “3‘ <4“ = 7E (-QuflJ—a Lsx kn = 'L—‘ 9V9. {W73 |$ flaunt“ a»). 2m \anao . so 11M kL‘tefndnkSWWS is‘h‘mhm mm» @QQ“9‘I 5° ls'Sn! 1’ bn-H F“ “a” bun: b1. 5 ‘L' s .L 4-! 24);;59 F‘Wn-‘L b 3b ‘J... v‘ 4" -' g-La 2‘ 3 4‘. 120‘19-9 SO'FwnaZ’ \s-snlcfia $1:J.-_L Friday, March 12, 2010 Calculus 1501B Page 11 Second Midterm Examination 2 11. (a) Define what it means for Zen to be conditionally convergent. mar/cs “=1 2, a... Is Mthwkx cram-3a.): l? i a“ W cm»: mm, bw’c mt- oJu-setdcdy mvwag‘k 8 (b) Determine whether the series marks 0° n Inn 12(4) 7 is conditionally convergent, absolutely convergent or divergent. Justify your answer using one or more appropriate tests. lei: Mtg: ma. My): ‘2‘?”- TIMLSI (J- \S MWWg om ie,“3 3" Kidd iS flimmrvxg Coy flag Rim ‘9.“ -: 9AM £44 :4; JQLM '/.,, haw n—aao h K'Vao “" 1 0 TM i Multan WW; Ioij Y\ MA-amahhg SéflPS Te“ , Mg] Zlqfl dwcvjeg b3 ’i’ka Coqumgom T551.— Ig.) EQK IS he.“ «Rudd». Canvaraew‘r. Hence. 70M \8 amhflbmflkx Calculus 15018 Friday, March 12, 2010 Second Midterm Examination Page 12 8 12. Determine whether the series marks 0° 11 ("U2 “21F” +1 is conditionally convergent, absolutely convergent or divergent. Justify your answer using one or more appropriate tests. [at Q“ =L_,DA-H (n3): (3'93 ‘43:}: c-~>"‘~m~+»r 1‘ @m), (-0“ L“ P) ‘ 42m =(LMDHI. (gm (“)1 Lam-HQ! = M. 4—— (2M9 (2M1) é\¢-O=¢ Tins, Zn“ ts «Stain-iota emvwzwk- bu) ’hu RdfioTfl'i'. ...
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This note was uploaded on 07/15/2010 for the course CALCULUS 1501 taught by Professor Shafikov during the Winter '10 term at UWO.

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Midterm 2 - Hiday, March 12, 2010 Calculus 1501B Page 1...

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