11Fe2s - _________________.___—— Prof. Girardi Math 142...

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Unformatted text preview: _________________.___—— Prof. Girardi Math 142 Fall 2011 10.18.11 Exam 2 NAME: K C¥ PIN: INSTRUCTIONS: (1) To receive credit you must: (a) work in a logical fashion, show all your work, indicate your reasoning; no credit will be given for an answer that just appears; such explanations help with partial credit (b) if a line/ box is provided, then: — show you work BELOW the line/box — put your answer on/in the line/ box (c) if no such line/box is provided, then box your answer (2) The MARK Box indicates the problems along with their points. Check that your copy of the exam has all of the problems. (3) You may not use a calculator, books, personal notes. (4) During this exam, do not leave your seat unless you have permission. If you have a question, raise your hand. When you finish: turn your exam over, put your pencil down, and raise your hand. (5) If you do not make at least 17.5 out of the 35 points on Problem 1, then your score for the entire exam will be whatever you made on Problem 1. (6) This exam covers (from Calculus by Stewart, 6th ed., ET): 11.2—11.7 . Honor Code Statement I understand that it is the responsibility of every member of the Carolina community to uphold and maintain the University of South Carolina’s Honor Code. As a Carolinian, I certify that I have neither given nor received unauthorized aid on this exam. I understand that if it is determined that I used any unauthorized assistance or otherwise violated the University’s Honor Code then I will receive a failing grade for this course and be referred to the academic Dean and the Office of Academic Integrity for additional disciplinary actions. Furthermore, I have not only read but will also follow the above Instructions. Signature : C Source VPTHLQ 51a“ 02 4&— ‘ él 7— 1' Fill-in-the blanks/boxes. All series 2 are understood to be n=1 1a. 1b. lc. 1d. 1e. 1f. 1h. nth-term test for an arbitrary series 2 an. If limnqm an 75 0 or limnnoo an does not exist, then 2a,; Cl ll/dra e 5 . Geometric Series where —oo < r < 00. The series 2 r" «A. 2:1. 0 converges if and only if lrl o diverges if and only if lrl p-series where 0 < p < 00. The series % o converges if and only if p > 4 2 1L o diverges if and only if p L Integral Test for a. positive-termed series 2 an where a." 2 0. Let f: [1,oo) —) IR be so that 'an=f(_n_ )foreachnEN "'JV/nor-anreasmj o fisa pOSl‘t‘iVC gflfl function (LS also 0&2 o f isa reajin function 0 fisa. COYH'MIJQLLS function. co Then 2a,. converges if and only if S 1 £02 dz converges. Comparison Test for a positive—termed series 2 an where an 2 0. (Fill in the blanks with a" and/or 1;...) o If 0 s an S bn for all n E N and 2 bn converge, then 2 an converge. o If 0 S b" S a." for all n 6 N and 2 bp diverge, then 2 an diverge. Limit Comparison Test for a positive—termed series 2a,. Where an 2 0. Let bn > 0 and limn_,oo fig = L. b If 0 < L < 00 then 20,; converges if and only if Z I“ comm rg¢ 5 ——5 . Ratio and Root Tests for arbitrary-termed series 2 an with —00 < an < 00. gm P=1imn~oo Let p = lirIin_.oo a" or o If p < \ then 2a,, converges absolutely. o If p > l then 2a” diverges. o If p —'- I then the test is inconclusive. Alternating Series Test for an alternating series E(—l)"‘afl where an > 0 for each n E N. If > an+1 for each n E N o limnnoo an = D then Z(—1)"an COMBO} {.5 can 1i. By definition, for an arbitrary series 2 an, (fill in the blanks with converges or diverges). «- 2 an is absolutely convergent if and only if 2 Ian] C 0 7W ‘5 I" C S 0 2a,, is conditionally convergent if and only if 2 a,, CO “Verfafiand E |a.,,| Adages 0 2a,. is divergent if and only if 2a,, diverg 85 1j. Fill in the 3 blank lines (with absolutely convergent, conditional convergent, or divergent) on the following FLOW CHART for class used to determine if a series 22:17 (111 is: absolutely convergent, conditional convergent, or divergent. Does Z |a,,| converge? Since Ianl Z 0, use a positive term test: integral test, CT, LCT, ratio/root test. Is 2a,. an alternat- ing series? ifYES 1} Does 2 an satisfy the conditions of the Al- ternating Series Test? ZanisMMMgAfi 1k. Circle T if the statement is TRUE. Circle F if the statement if FALSE. l T I F If 2 an converges, then lin1,,_.,.,‘D a1, = 0. T CF) If lining“, an = 0, then 2 an converges @ F If 2 an converges and 2 bn converge, then 2(an + b,,) converges. If 2(0.n + bn) converges, then 2 an converges and 2 bn converge. N _ N T ® If SN = Zr", then SN = r T , for when r aé 1 sincewedon't llkato dividebyurc 1 — 'r 1L=l Source. a, Warm w}? "(Ldvxls'olch hull-almle T‘Ol’lflm. ’/ 2. Check the correct box and then indicate your reasoning below. Specifically specify what test(s) you are using. A correctly checked box without appropriate explanation will receive no points. E absolutely convergent °° (-1)" 2 D conditionally convergent '5, L4) n l A/ ? 2’: >l Zl nL Z n C/o’vw. Since. WV; 2: (:22 AL» alas.ch SW SLWFIhMMoMA‘p/(Lm 3. Check the correct box and then indicate your reasoning below. Specifically specify what test(s) you are using. A correctly checked box without appropriate explanation will receive no points. [I absolutely convergent [Z] divergent You may use, without showing, that for n 2 1 4n7—n6+2l big, RN; 5,, PaginVc—n‘umol sane; _ 4n7—n6+21 “n-=m>°' Hint. What would be easier: CT or LOT? '4 71mm Liv—aw "5% ill, , LL 1,; __________..... , Lew1L Ilng—n’rn- ”’ “A? “ .L. 30 by» Jaw LCT “’1 En: n 1r (1 C6 vtlln .300 Ll a“ Wn—n+ll.flii;fl”.br L H g_{\+r’+ I IerLnH'l' 0V vL . Comma/ES 6n, it 3. m caflwllaflo‘ms ow +114, I'D/IOVViHj W w 4 m, j..- < “Min 4—2! Hm ‘ 'lng—n+l1 n’-’ W 4n?’nb+1l Lf q E —— <7;> We 2 Abba. q ’- < “7,\ Mn 45-) ‘+(Hn<3-'nkl?) < H“ (4h7—hbi’1‘3 <=> an—Llwrw 4 L’L‘ng "'Y‘? ‘“ 33‘“ q. 4.)} 0 < 'lln man»? but 7900 : "Hvx +235ex'é8 Vu)= ’11m“~+23E __ 5 \c'ogzo es» 'Xa « Z” if; 3 Li $30 _,(__,_‘C_,,<£_. ,. x0 #(xo) ’3 Ilw M) ’ '°° 7v?!) #0) =- '59 HQ) :: —-;z)o(, / 4 4 Jew—{:13 -292), ’2 m, H(2)‘3-1+|—+ ’ agar '” 4b. Check the correct box and then indicate your reasoning below. Specifically specify what test(s) you are using. A correctly checked box without appropriate explanation will receive no points. ,3 absolutely convergent a: (—1)n (Mn—l 1)! [:l conditionally convergent D divergent (nHll _ (n+0 ! (ml Cougar Mame-s: ” MT”?- = M. “P (1 (2M2. ll- (ZnH). ,i—P—"w‘d'fl/f l(n+l LZW' . l _ . ,__,_——————- amt : (“MN .(ln—lfl i -: Ely—’2 ' (“val (2n)(2n+|) "If m “HT” ’ M. (ml)! “‘- ‘ 0 (MI) n+\ fl %%5:0=€<\ :: : / 7' 2.; M Lin“ Jan 1 Ll“ n L 5. Check the correct box and then indicate your reasoning below. Specifically specify what test(s) you are using. A correctly checked box without appropriate explanation will receive no points. {:I absolutely convergent El conditionally convergent ll-WL 11L...— :: ‘75 H.900 3n1+g“,\ livl AV vow. 237?:— 3' Solvfinflkwwultvll” 3’ 3"l in m“ 3n HZn l “$00 n “Lkl Hmwfig 'PFOIOILM g “57’ 44; I3 6. Check the correct box and then indicate your reasoning below. Specifically specify what test(s) you are using. A correctly checked box without appropriate explanation will receive no points. D absolutely convergent 00 I Z (—1)" H” E conditionally convergent n=1 V n D divergent J Lu: , 2—~ln 19. Lctf(:r)=—,_-.Thcnf(z)= I<thenlnz>2orz>e2,sofl m I [,2 fl is decreasing fem > 8. . - , . Inn . 1/ I BylHospxtalsRulc 11m — = n — lim 2 ‘ m In” “M” n” x _ __ = 0' m _ "_ - ‘6; a: 1/ (2 v/E) mm «a so 653168,§l( 1) J; converges bytnc Alternating Series Test Limit Comparison Test Let b“ > 0 and link—0° if = L. his ‘ (LCT) [f 0 < L < co, then [ E a." conv. ==~ :bn conv. ] (you DO need to memonze I one: M 1H; 2 I If L = 0, then [ :bn conv. = S on conv. ] (you do not have to memorize this one: / ,4 If L = 00. then [ ‘7‘ b... divg. = E a." divg. ] (you do not have to memorize tins one) . I. Let in = L um :19“ 3ng Ha. p—iews) Pwfi l, 8 unfliflAcfl, EOMI/OJ’ be Sum 31 it? 31mg 90/0 7. Geometric Series. (On this page, you should do basic algebra but you do NOT have to do any fancy arithmetic (eg, just making up numbers, you can leave @9171 as just that.) Let, for N 2 51, 3n+1 N ans/=2:257I . n=51 7a. Do some algebra to write 3N as 2§=51 c r" for an appropriate constant c and ratio r. 7b. Using the method from class (rather than some formula), find an expression for 3N in closed form (i.e. Without a summation 2 sign nor some dots .). ...
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11Fe2s - _________________.___—— Prof. Girardi Math 142...

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