10sE2s - Prof. Girardi Math 142 Spring 2010 03.18.10 Exafi...

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Unformatted text preview: Prof. Girardi Math 142 Spring 2010 03.18.10 Exafi MARK BOX PROBLEM POINTS 1 a — j 30 _.__¢' - " "2==:.:%. r-ixq‘: (1) To receive credit you must: Xxx” (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 lin‘e/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. 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 (from Calculus by Stewart, 611 ed, ET): 11.2~11.8 . Problem Inspiration: Mostly homework and old exam problems. See the solution key for details. 00 1' Fill-in-the blanks / boxes. All series 2 are understood to be n21 Hint: I do NOT want to see the words absolute nor conditional on this page! 1a. nth-term test for an arbitrary series 2 an. If limnnoo an 7E O or lirnn_>00 an does not exist, then 2 an Cl We"? 9? 1b. Geometric Series where —-oo < r < 00. The series 2 r” o converges if and only if lrl < i o diverges if and only if {TI 2 l 1c. p—series where O < p < 00. The series 2 n—lz; 7 l . . , < I o diverges if and only if p — o converges if and only if p 1d. Integral Test for a positive—termed series 2 an where an 2 0. Let f: [1, co) —> R be so that ,an=f(f\ )foreachnEN o f is a funcmon . f is a (33; its??? function . j .7 4 Then an converges if and only if i. TU )“X A” converges. 1e. Comparison Test for a positive-termed series 2 an where an 2 0. 0 HO 3 an 3 bn for alln E Nanden Carve-‘98 ‘4'" ,then'Zan Cm 7» oIfOSbnganforallnENanden diwaffiev ,thenZan ‘7'- a/CWfi 63 1f. Limit Comparison Test for a positive-termed series 2 an where an 2 0. Let bn > 0 and lirn,H00 = L. \ 1“ {wk 1 If Q___ < L < G3 . 7 then 2 an converges if and only if . 1g. Ratio and Root Tests for a positive—termed series 2 an where an 2 0. Let p = limnnoo “2:1 or p = limnnoo (an)%. o If p C l then 2 an converges. 0‘ If p 7 i then 2 an diverges. o If p : 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 > an+1 for each n E N o lirnnnoo an = 0 then Z(—1)”an is "V a" ‘3:- ‘3 3‘ 1i. By definition, for an arbitrary series 2 an, (fill in the blanks with converges or diverges). 0 2 an is absolutely convergent if and only if 2 |anl COMET-$8? 0 2 an is conditionally convergent if and only if 2 an CQW‘PWQT and 2 ion] '3 3 0 2 an is divergent if and only if 2 an lj. 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 2701:” an is: absolutely convergent, conditional convergent, or divergent. Does Z [an| converge? if No Does => if NO 2a,, is limnnmlan[=0? => ‘ Since {an} 2 0, use a positive term test: {an WW6” integral test, CT, LCT, ratio/root test. if YES ll - if YES ll Ea is madden Cygr‘v'é‘lfjggz‘l Is 2% an alternat- TL _—____________— ing series? if YES ll Does 2 an satisfy the conditions of the Al- ternating Series Test? if YES ll 2. Circle T if the statement is TRUE. Circle F if the statement if FALSE. T ® If lirnnnoo an = 0, then 2 an converges (T) F If 2 an converges, then limnnoo (1,, = 0. @ F If 2 an converges and 2 b7, converge, then Em” + bn) converges. T {f F If 2(an + bn) converges, then 2 an converges and 2 bn converge. K. 1/ _ N 7‘ _ TN-l-l 1 _ n _ F IfSN—Zr,thenSN——-1——_—r—. ‘E n21 ‘4} 4W“ "3 .1," ~. _ €- h} 2 {- fl; \, , ,9 , g It} 2; d K l ’ WMaWMwwm-w wmmwm’““m‘ 3. Check the Correct box and then indicate your reasoning below. Specifically specify What tes:t(s) you are using. A correctly checked box Without appropriate explanation will receive no points, W convergent °° (-1)” Z n E conditionally convergent I M"‘ \. pie!” w" 1" \ - ~— l \‘1 i ?\ PVT“, rm A‘s”? ’i—x {A , L 7 “‘ :Uh+\ r- h 'er, h V (“Kali R M (f C31 m 2:20 sow W Z n r‘ .2” F4303 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 00 1 Z ————-—————— D conditionally convergent 71:1 \/ n (n+1) (n+2) \ D divergent on 1 W i Z s 2‘ If; ,3 :qh b'—\ \ er-ixflrlfi it ' “oi: LCT \ bf“ " K “’5\ «‘3 ‘ _ ’5 p 5/ 1 i { r\. “9“” a? “kw J ' l oz ‘r V0 l i m ,f , (’59 a” \ \>“u< V ‘3’ K iv)" bf)...” CkLI‘l-k \r4333 3 - - w, (j r - "w. ---- W“, W “0‘ J a .1 " ‘2 : xv“? M E E 3 2 lflr ‘ 2:; 53‘)" ,, " Fab.“ f ‘ or V V a 'if” ' ‘ .3,» i \,, , K T WWW t 912*? (of '7‘ , ’5 _ q, ,3, ‘ J” r { g m » l , . C_,~’;r‘~§ 5 -’ (C kg ‘3!“ “" ‘ € 9395 \fr {r “Blend 5. Let n! (2n — 1)! an: 5a. Find an expression for “2” that does NOT have a fractorial sign (that is a ! Sign) in it. TL 5b. Check the correct box and then indicate your reasoning below. Specifically specify What tesc(s) you are using. A correctly checked box Without appropriate explanation will receive no points. absolutely convergent 6. Consider the formal power series n n=1 Hint: (5m + 10)“ = [5 (m + 2)]n = 5n (cc + 2)” = 5” (a: — (—2) )n - j, The center is $0 = ‘5 ‘ and the radius of convergence is R = 5 __ . 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. (Ar hulk-gratin A -_ I I. ; ‘ . _ f .1» (it 1.1 Wim- a» 1 x w. a”: 32m “ v : W, E! i t ' C ,4 \ r” l v Y ’A (I) ‘3, .s i A. . . f1: : z E ' f “ ‘ 1‘ u 1 I .1 r E ‘ {I a); 4 h L f h ‘3 A‘ v r l s .. : % ‘», {A “ l l ' : ‘ “ l f “WW 3. a: r r. . ., 5 _ . g“ i Mi ....... u I [6 wwwmt K C30.» 4—H“ a} m n ‘2‘ l i kn": w 3+ .4; A r». w EAL/ii, {La ; . x _ A ~ I; ; W t,” g f ?'\ ‘wl " : (“*2 g: i at i )2. M!) V: \-l u‘ " ‘l V} i .... , r »* ”" ”” f r , [l “K 'x Wax; x ’ i 1” f V3 E, a, J a V I“ 7. Consider the formal power series i W Tl TF2 (1n n) Hint 1: (1%??- = so would you rather use the root test or the ratio test? Hint 2: ln(ar) rln(a) but (in (a))r # rln(a) + The center is 1130 = O and the radius of convergence is R = Cm 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. (j, \30 A} i F L 'Q lv\ (7:) ”r yrs {,«c 3 5»: g" A 2f _, ,. n/ v / Eng—r {a} a.» V lE r/ X h l I“, H“ VI \ii n‘ N 1 ) O : I) J? 1’1“ _..__ h F fl’l if” b Egg? “E F b Fill—in the 6 blanks. Consider the power series 00 Z (—1)” anxn n=1 where all of the an’s are positive. Let‘s say that you know that if if if 0<zc<17 $217 17<$ Then this power series has: center at 3:0 = Also, what can you say about the following interval? Fill in the blanks below with: o is absolutely convergent o is conditionally convergent is divergent then :X—l)”L anxn' converges then Z(—l)” anx" conditionally converges then 2&1)” aux” diverges . l7 and radius of convergence R = inconclusive (not enough information given to decide in general). : 2 g , {if if - 17 < m < 0 then 2H)” It a if x < —17 then Z(_1)n anmn , a r 1f ‘xZO then Zl-Dn arm” 0mm” H“ <4“ Vela?" m V .5," ,'~’€L»n€_ he , if a: = ——17 then Z(—1)” 0mm” (i i i J , s h ‘ I Z w; I : fl W \ 5‘ P 2: { i; any”) 9. Geometric Series. Let, for N _>_ 102, N 2” SN 2 Z 37n+1 ' 712102 9a. Do some algebra to write 5 N as 25:102 c r" for an appropriate constant c and ratio r. 9b. Using the method from class (rather than some formula), find an expression for 5N in closed form (i.e. Without a summation 2 Sign nor some dots 9c. Does 22:102 3% converge or diverge? If it converges, find its sum. Justify your answer. ...
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10sE2s - Prof. Girardi Math 142 Spring 2010 03.18.10 Exafi...

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