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Exercicios resolvidos James Stewart vol. 2 7ª ed - ingles

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88 0 CHAPTER 11 INFINITE SEQUENCES AND SERIES<br />

• oo x2n+l 00 x2n+ 4<br />

51 . arctanx = 2: (- 1)" - 2 1<br />

for lxl < 1, so x 3 arctanx = 2: ( -1)"- - for Jxl < 1 and<br />

n = O n + n =O 2n+ 1<br />

I<br />

oo<br />

x2n+5<br />

x 3 arctanx dx = C + ,..;;:o(-1)"' ( 2<br />

n + 1<br />

)( 2<br />

n + )' 5<br />

Since ~ < 1, we have<br />

(1/2) 5 (1/2r (1/ 2) 9 · · {1/ 2) 11<br />

1.5 - ~ + ~ ~ 0.0059 and subtracting Til ~ 6.3 x 10- 6 does not affect the fourth decimal place,<br />

so J112 0 x<br />

3 arctan x dx ~ 0.0059 by the Alternating Series Estimation Theorem.'<br />

(1/ 2) I oo ( 1/ 2) x4n+1<br />

n =O n , n =O n 4n + .1<br />

53. v'1 + x4 = (1 +.x 4 ) 1 1 2 00<br />

= 2: (x 4 )", so }1 + x 4 dx = C + L: --and hence, since 0.4 < 1,<br />

we have<br />

1<br />

I= } 1 +x 4 00<br />

0.4<br />

(1/ 2) (0.4) 4n+l<br />

dx = E<br />

o n = O n 4n + 1<br />

(o.4) 1 ~ (o.4) 5 H-4) (o.4) 9 H-t)(-~) (o.4) 13 H-4)(-~)(-~) (o.4) 17<br />

=(1)or+l!-5-+-2-,---9-+ 3! ~ + 4! ---rr- +· ··<br />

(0.4) 5 (0.4) 0 (0.4) 13 5(0.4) 17<br />

= 0.4 + ----w- - ----;:;2 + 208 - 2176 + ...<br />

Now (O~~)o ~ 3.6 x 10- 6 < 5 x 10- 6 , so by the Alternating Series Esti~ation Theorem, I ~ 0.4 + (0~~) 5 ~ 0.40102<br />

(correct to five decimal places).<br />

x -ln(1 + x) x- (x - l x 2 + l x 3 - lx 4 + lx 5 - • • ·) lx 2 - lx 3 + l x 4 - l x 5 + · · ·<br />

55. lim = lim 2 3 ., 4 fi = lim 2 :l 4 5<br />

x-0 x2 x-.0 x· .,_,o X 2<br />

= lim(! - l x + lx 2 - ix3 +: .. ) = !<br />

m-+0 2 3 4 5 2<br />

since power series are continuous functions.<br />

sinx- x + l x 3<br />

6<br />

57. lim<br />

x-o x5<br />

since power series are continuous functions.<br />

. 2 ~ ~ ~ ~ ~<br />

_59. From Equation 1 I, we have e-x = 1 - I + - 21<br />

- I + · · · and we know that cos x . = 1 - - 21<br />

+ - 41<br />

- · · · from<br />

1. . 3. . .<br />

Equation 16. Therefore, e-"' 2 cosx = (1 - x 2 + ~x 4 - ·- ·) (1 - ~x 2 + i4x 4 - - - • ). Writing only the tenns with -<br />

degree < 4 we get e-"' 2 cos x = 1 - lx 2 + .l.x 4 - x 2 + lx" + lx 4 + . ·. = 1 - !!.x 2 + ~x 4 +- ·.<br />

- ' 2 24 2 2 2 24 .<br />

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