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MA241_test_4_soln

MA241_test_4_soln - (When you are ﬁnished fold the test...

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Unformatted text preview: (When you are ﬁnished, fold the test ovcr vertically and write your name on the back) MA241—012, Fall 2008 Test 4 No calculators are allowed on this test. Notes or other aids are also not allowed on this test. You must show all work and sufficiently justify all steps in order to receive credit for a problem. Be sure to state which convergence tcst(s) you are using and show that any conditions on the terms are satisﬁed. Answer questions in the space provided if possible; you may continue your work on scratch paper if needed, but clearly indicate which problem the work is for. 1. (6 pts. each, 18 pts. total) Determine whether the following series are divergent or convergent. If convergent, ﬁnd the sum of the series (i.e., what it converges to). I 00 n 0‘7 n’l 5” "' @2363) 23:74am?) 42%? we“ '1" bv—J uw 11:1 A )(‘l (l 2? 3‘: "" "2 ‘ 7‘ r ‘ 7 - _ _’.l f0 I ' C? \$9 feta) = (1' i) +(:' 7%) Hi p 540 .3. e rfc’f caﬂl/ﬂf/n 7 9 5 I J ' )3 [/7 5 Z + «x. ;a W W W i at 00 1 5 11—1 ({4 — (C) ,3 A jeep/taint \$52,146; W, a- 2. (8 pts. each, 56 pts. total) Determine whether the following series are divergent or convergent. Use each of the following tests once (one per problem). Show all details. 0 Integral or Ratio Alternating Series Limit Comparison (with a p—series) Direct Comparison (with a geometric series) Absolute Convergence (plus a Direct Comparison with a p—series) g0 an {Lao} =9 0/66. 23' A 5011 VIC/yew?!" ,9 '5L“‘v"/5’7J __,—— 47/50 con/I ye! OnthTerm oRatio °° 3"*1+2 pa 3"" 0° /(3)n” A ’ / z " ’ who ISA card 56 E 3M”? / at as / —’—"’ IL/g'i’ . So if yin/6765J 50 [y DC ) ml an A 50 i 2 0° (dun—1n n - -——' '5 a5“ (x): w/CX 1 AS/ (b) Z n2+1 An 5/1,”, ( /(x}-X,H I p ) 7F AMI) 71:] M l/sz‘ 65 J //WI n _ a 50 I, A57”) U nsz con , an n-aao 57*? _ I i '4 0° () h Sinm) E ’ MAMA smn \< ___§_ ACT (C) E n3 n4; / n3 n“ n ‘96 57340“) W SMO‘V 5 5 é Z 3 go ) r155]! COﬂVB7€ ) o i J [1:] ﬂ / e)“, 0° _ n n+1 , ﬁ/r _ 3 (did (d):(1+ nﬁl/(Wyr' " _ = M a </ n=1 M So ly LL/xe I’mfrﬁ 535%; A; n! com/975’)“ Q 00 ('6’. Zita/m1 (e) Z 716‘" Jnﬁ7ra/ fa): XC—X ,5 conz‘ ouan f(x)= e'x—x€~x= eYI-X) <® 6" n21 r {I — - - - x 0 \$1 0 [I w) 50 /(x) ,5 466 jxexalx ____. 'X€X+ 6,, 4x;_mx_a , e {—l’x) 1 z w; #:630 as X W -e a 5 ofxg‘xXx = /I‘r"1 / X) ’JL‘: /:'M C-/ + 0234 [M (4-0/9 [M -1 3 o I 6‘ 63 , game e I {am:f inmg=0 few W _ W — . 50 fxéxalx com/(’79:) so A; 613 ll-Iﬁlifd/ 56575) ﬁne" Z4/56, (om/5765. 0' ﬂ?’ €14! ’ /IM ﬂ- : J, a I ‘A I Z4290 ' n/gwﬂo /eM' 7‘ / - new 61/) 5 </; ‘0 5/ ﬁe rat/W \$651}, Eng can a L67— (f) f: _3”_ ‘ n=1 m M 3 W w n ‘90 I “ r I . asa (—57— W’Ltl‘ 5 «TH ; “:5, 74' (or rg} W a; ’4 I ' ?4 4 1 M r» J7“ Liz—.— = We) #w new n‘fffﬁ‘f’ : ‘ 7 v- 3 - Ill-l +§n 4" 3 ) ) ~17" n-éM ﬂ4+ﬂ{/ n M ’1 d r E '42—,— 4/50 aim/cf £35 ,0 5/ ACT‘, Since g n “3/8", W, 1iﬂ4+yﬂH I 1 if f: n2+1 1 / :1 K) 'f' :L 7;,«1 (g) *2—— /""” ; 4 14h ’4 n:13n “a” 3n? +6ﬂ+f 3 5d 53’ I4 LLB/“M fcffj 3. (7 pts. each, 21 pts. total) Find both the interval of convergence and the radius of convergence for each of the following power series: I F 00 n /: 2 < (a) Z % jag “P MU!” éejf “ name; X” nac- n-H X/ /X/ / 11:} when '/ ( x 4/ I M A M ’1— 3 a -5€/‘f£’5 Niki» [2‘ I, ) yo I’L‘ 461/575); Clack Ki/ Z .l/- : Z n71 I F W "1’ ‘0” AC, ’ v IOC. ‘ 50 W3 ﬁle) "at IQCMQ / )7) ‘ c , ’ ’ﬂ 4: " 5y Jed/MSW, IO :‘ < y; I k ——x I E r?” 55¢"; v M2 W’ 4m“ m «‘7 ' n w ' n:/ «lm 1 yo kw" er (’5 L; ﬂjf‘ 50 mfﬂO/Vl’é hm J— ;@ o c: F?” {WV (7 Arm! "—9.79 «FT 1 n/ I" x=-/ m 5459 IOC =/ 59) \$04 = [—/} I) aim-3 f3 . b '2 —1" n+1 ()7;1 n( cc ) /M (MUMUx’I) /_ /m /(ﬂ+.)(1x+l)/—>m I I f ‘ {-4 be 5657f new /,,; (Xx-I)” new W1 /€55 02x +/ w [email protected] I06=Zrill / . M M/: /;M ﬁraM/(Mbe7/X. y /I' 1",” ’ 'ft’ fat up raéfa 655'5' rte/:6 hr”)? x"r 50/ foc=('po)oo)) 2:00. 4. (5 pts.) Use a geometric series to express this number as a ratio of integers (i.e., as a fraction): 0.—6=0.363636... = ,36 + . 0036 +«000035 1 0" gs / n—/ 36 c 36 _L 2.6. —’— +_‘, = _._ ~— : ’— - z :12. + I; I”) f’ [w my ":2, we (/00) 1.0") r 71 4 ’75 M l"7</ Bonus 1 “ if |r| < 1, and diverges if |r| 2 1. converges to 1_T DC (5 pts.) Prove that the geometric series 2 a ~ 1""— n=1 (Note: I did this proof in class, and it’s also in the book). \l ...
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MA241_test_4_soln - (When you are ﬁnished fold the test...

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