09Fe2s - Prof. Girardi Math 142 Fall 2009 11.12.09 Exam 2...

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Unformatted text preview: Prof. Girardi Math 142 Fall 2009 11.12.09 Exam 2 MARK BOX PROBLEM I _ NAME: 0 7 m CLASS PIN: O 0—3L 5—1 CD 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 b0x 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. If you have a question, raise your hand. When you finish: turn your exam over, put your pencil down, and raise your hand. (5) This exam covers (frorn Calculus by AntOn, Bivens, Davis 8"h ed.): (a) SectiOns 10.1 - 10.6, 10.8 for the inclass problems (b) whole of Ch 10 for in class fill-in-blank and true/false problems (c) Section 10.7. 10.9, 10.10 for the take home part. /Problem Inspiration: See the answer key. 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. Furthermore, I have not only read but will also follow the above Instructions. Signature: I :‘V a 1 J an ad O E J E (Y r I 1.! 1&2. 9P“? “‘5 " ‘1. {a Li “,5; is 3 Spa-sift: {7 */ pill" J :F l” " u 7‘)“. all? i" __ r- SUI: _ — E” HF f" S Hka I9” "k "'3 ‘ ’ "rah ~ 00 1- Fill-imthe blanks/boxes. All series 2 are understood to be n=1 Hint: I do NOT want to see the words absolute nor conditional on this page! 1a. Sequences Let —00 < r < oo. (Fill-in—the blanks with exists or does not exist, tie. DNE) o If |r| <1,then limnnoorn 890541;- . If Ir] > 1, then limnnw 1"” {W E o If r = 1, then ljmnnoo r“ (mast s . If r z -1, then 11111,...on 1"” DNS; 1b. Geometric Series where -oo < r < 00. The series Zr” 0 converges if and only if lrl < i o diverges if and only if |r| 3' l 1c. p—series Where 0 < p < 00. The series 2 13—), o converges if and only if p 3“ 5 o diverges if and only if p 5 I 1d. Integral Test for a positive-termed series 2 an where nn 2 0. Let f: [1,oo) —> R be so that oanzfi 7“ lforeachnEN o f is a Dcsiiiu’fx function e f is a If :T'Ifli a LA 9MS function 4. f is a clacrsasins or not! iflCrfisirldi) function . Then 2a,, converges if and only if 3:0 1:6“) converges. 1e. Comparison Test for a positive-termed series 2 on where an 2 0. o IfUSo;n SbnforallnENanden ¢<‘3‘"-"-'3Ff?x‘.$ ,thenZo.n (“'GnV'CV‘E-S . o HO S hu 3 as:n for all n E N and 25,; $1? I"?! , then 2 on do”? a 55 . _d If. Limit Comparison Test for a positive-termed series Ea... where an 2 0. Let b” > 0 and 1i1'r1,3_,c>° ff: 2 L. If 0 < L < ‘30 , then So.“ converges if and only if Z 1'31s ‘15“ V" W? 5 . lg. Ratio and Root Tests for a positive-termed seriw Z: on where an 2 0. ‘ 1 Let p : hm”.H30 “3:1 or p : Brande.o (an) 3. o If p 4 1 then Ea... converges. o If p ‘> l then 2 an diverges. a If p T— ! then the test is inconclusive. 1h. Alternating Series Test for an alternating series Z:(—1)“‘an where an > 0 for each n E N. If ' a“ >— an+1f0r each n e N (deer-Casi n3) . lim-fl—boo on = 0— then :X-llnan CUn vars: {,5 1i. nth—term test for an arbitrary series 2 an. If lining“, an 75 0 or limnnm an does not exist, then 2a,, Ali '1! 6 (iii l" 1j. By definition, for an arbitrary series 2 an, (fill in the blanks with converges or diverges). 0 2a,, is absolutely convergent if and only if 2 ' OH ‘4' \‘1 I"? €— 5 0 2a,, is conditionally convergent if and only if 2 an Ear. fires; and Z |anl oii‘l “I f1 r. S 0 2a,, is divergent if and only if 2a,, “9.! SHE .93 — U 11:. If a power series in :r — so has radius of convergence R where 0 < R < oo, then the power series is: o absolutely convergent for “X. E. ( xo " R ’X a i" R) odivergentfor 9143(- ‘R. anci— “Xx-R<’X . Let y = f(:r:) be a function With derivatives of all orders in an interval I containing $0. Let y = pN(:c) be the Nth-order Taylor polynomial of y = f(x) about :30. Let y : RN (3:) be the N tlrl-order Taylor remainder of y : flan) about 2:0. Let y : poem) be the Taylor series of y : f(a:) about 3:0. 11. In Open form (i.e., with and without a Z-sign) ff.“ 1p. Did you write your PIN on the cover page (under your name)? It’s worth 5 points. ' U 2. Circle T if the statement is TRUE. Circle F if the statement if FALSE. To be more Specific: circle T if the statement is always true and circle F if the statement is NOT always true. Scoring: 2 pts for a correct answer, 1 pt for a blank answer, 0 pts for an incorrect answer. T ® If a sequence {an}§f;1 satisfies that limn_,oo an : L and K H f : [0, 00) —n- R is a function satisfying that f(n) 2 on for each natural number FER n , g x “I ‘ then limmqoof(:c)=L. Cmgdfif ,Lfiy) ‘:: Slh (ZFK‘i primed I } (9 F If a function f : [0, co) —> R satisfies that linion f(a:) :. L and {an}:°=1 is a sequence satisfying that fin) : can for each natural number n, then lining“, on = L. T :1" W T F If 2 an converges and Eb” converge, then 2011; + bu) converges. I L d «L T r' F If Emu + b“) converges, then Ea” converges and 2b,; coriverge. {l 1 Ta n’ 'L N - h ) r17 _ rN‘i-l Cr) F IfrgélandSN:Zr”,thenSN=fT—foreachN>17. / NOTICE, the above sum starts at n : 17, not at n = 0. M n 18 + 1" -- 1 r 'i" - xi SN F a 4r \A fl Jr (“H “Mm”: ii"; '_ i” r ' 1“ SJ 1 LII—f "— r: - fl —, {Age} 5—1“ 34:1 r ‘ I 3. For the following SEQUENCES: o if the limit exists, find it o if the limit dew not exist, then say that it DNE. Put your ANSWER IN the box and Show your WORK BELOW the box. mm (471 + 1)(5n + 2) z n—voo 17502 13m “yaw = E; n—roo 177.32 ti :1 ljm (0.9999999917)” : n—poo 4. Check the correct box and then indicate your reasoning below. Specifically specify what test(s) you are using. A correctly checked box without appropriate explanation will receive no points. |:| absolutely convergent n 3' J ?-fii..;5 1:113, < l :25 CL..@L"%L$ I So “0% M\) .3 | . f". _..-I I “W.” A . ‘1?“ I;- X/. _ )l [m x & ‘CrU—r 4:..- *' r----1 M-.. I”? “94 ’ln {3,011 01 ‘ GEL t . r -* - H ix i :m :1? r U ‘I - Y1 0C Edi n . 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. El absolutely convergent {fl \n ' Mo ‘17“ Pm“ fig—'2“) an“ haw ” 6. Check the correct box and then indicate your reasoning below. Specificafly specify what test(s) you are using. A correctly checked box Without appropriate explanation will receive no points. I: absolutely convergent Ho / Hint: For any 0 < q < co, ifn is big enough then Inn < n9 and so (tn—fir; < F11”) . x [i n (x i— l) " ( X ‘ J l 7. Consider the formal power series / // i (—1)“ (93+1)” ' n —l __ “=1 _ i and the radius of convergence is R : The center is x0 = As we did in class, make a number line indicating Where the power series is: absolutely convergent, conditionally convergent, and divergent. Indicate your reasoning and specifically Specify what test(s) you are using. Don’t forget to check the endpoints, if there are any. Avg {Wuiimi’rld ‘ ‘rt I \L/ ms‘ m \f/ _- J 1‘ ’2’ "I 0 n+1 ;. p, pull ixtll l-w; -: lel 1- {7x44} IL I f {ia'h‘L ‘5' “Jane h'rf .. Im _____#_ t f, , “5&0 y,“ l) “4"” m»; {XHJ r_ if __ 4 k 6'" . .‘xn lied lml W L,“ H l '4'" (1+ in 1““ “f ‘i “‘3‘” 7‘ Q -: [lib } j :5 “$00 “liq l H‘HL r“, , xiv! aloe. (low “M i l n N _J‘ ('0 (“ll :1 n“ L Add ~=~ 3"- C‘Q’i"; : Z r: ‘”' n” P'mms At } n p;- Lu AST VOHln (din 4r CNN“? d No ‘" [if z '73". .th-m iX-r ' n I Zciln 30' h 9&0» L913 m2n+1 0‘“ " (2n + 1)! that does NOT have a {rectorial sign (that is a I sign) in it. “n+1 Find an expression for -..—---"" _. —v-w“ _ '- "‘"‘_—_Fm U v-w—‘W'fl'w" -‘="'_"‘ -- “" \I 2.11"}! z: ‘-"" —-=--*-"'"" a f 'H I 1‘ f q n inf-I an {4,n\ }, )1 KLMQE 7L ,, 5- ‘ air-.571 2- M an I 2,. x ff?“ '1 I. * -- , s -v — ~ ~- gee—ee— " I n - — \ r '3 A J .1 I“ i , x2 71 '5' I-l / r J [I ‘I " Z (—1)“ (2n + 1): CK) The center is 5‘30 : 2 and the radius of convergence is R : As we did in class, make a number line indicating where the power series is: absolutely convergent, conditionally convergent, and divergent. Indicate your reasoning and Specifically specify what test(s) you are using. Don’t forget to check the endpoints, if there are any. abs. WLU/ . . flnfl __ 2 ‘ fl fl #5 ' W E . 4* l .— l|\\ 1' Ep- "a I. I“; “:- iive \ ._.—- _ Lop ‘ :9“ ‘1 -? liming; j AN- ...
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This note was uploaded on 12/13/2011 for the course MATH 142 taught by Professor Kustin during the Fall '11 term at South Carolina.

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09Fe2s - Prof. Girardi Math 142 Fall 2009 11.12.09 Exam 2...

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