sol10 - MATH 138 Calculus 2 Fall 2010 Solutions 10[a.1...

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Unformatted text preview: MATH 138 Calculus 2 Fall 2010 Solutions # 10 [a -_- .1 1 _.1 ,r. T 3 .r. ,3 l" .1 .r .1 .r . _ .- _ ’_ .r .r J i J HinJ': 2 [- — .l' . r] _! "“.bt'|hlllJ-.l~- _§?+Tfi?_fi+'nafld : — —— + — — _ n H [2” l I)! I” ‘l, f 1 ., _ .1 h'lllJ" _ -” 1 if" ii" ‘ . 5mm — J' _ . l i F J’ '_ _ l T '— —i + 7'5 E l Thu“ l'flh‘J-u— _ 9'3}. (‘ [1 * 120.1111!) * " (i' 1U?) _ 1_m” . 1fl(1—.-1;::“+fim‘-a‘r1‘°+'“) mus—e: '19}:on—[1+x+§lm3+§1z=+fir4+§x5+ézfi+nq a; m. r 2 _ If! is an odd fimclion. then J'(—:r) = —f{r) =:- ): {—!)"c,.m" = Z —c,,:r“. The coemcicms ofany pnwer series 1:: "“0 are uniquely delcmained (by Theorem 1 1.10.5). so (-4)" c" = —c,.. ”1'1 i5 even.thcn (—3)" = 1. 50 r?” = —c“ = 2r." = 0 => on = (1. Thus. all even coefficients are 0. that is‘ afl=c2=c4:n.=[)l at; w 2 N on '2“ no 1 n will I C‘cu an H I ' % ELI :’ I‘m)“: Elfin? :n§u%=.§ufim2 515.1,“le gum“ "‘ "~ WM as _ [2") 011 n} (______2n 11)! a I‘M byflm): Z fugfmar" Comparing cnefficiems for k— .— 21: we have f—-—(2n§!.— }— Frlz'l. => 2 (fl)=1——t—-! . k-—'[I ' 4; : Since f{"}{[l) = {n + ”L“ the Maclaurin series J10 07’” W is ;_ f_‘"lflw)" w (7: +1 ! ,, m . Zu— = Z “I } .‘1‘ = 2 (11+ l}.T". Applying the Ratio Test with a" = (n + 1].r” gives us !1'—- ”=0 - 11:0 ‘ fin-p 1 _ ‘ [11 + 2);::" H . n + '2 _. "111.120 T‘ — ‘32:; W = |:::| "151; n. + l = |.-r.| » ] = lnrl. l‘or convergence. we must have |:r.| < L so the radius ofcmwergence h’ = 1. at Cl: {hfll ) (n+l lln'll) :7 {21‘1”} §il\Il—;;V_L'i :1— n‘a‘A 31—390 “1'2 W) on n x no "—1 3 no :(11) E: :_ _ z_ .e—~1= I [a __1 = 8 _ n} =* 3 1_-:E:ffl => E :5: nl =5 I dz O‘FEE; n=0 “=1 “=1 "—1 2n—1 z= Ef-l)"-—- => m3-l=§(_l}n :2 mag-1-” "3 2n)! =1 =¢> z _ "21(4) (2")! ( ml (Twirl 1d 2 : —(1'.— a.) + a“, _ £03 ...
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