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

56 0 CHAPTER 11 INFINITE SEQUENCES AND SERIES
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63. :L: en:z' = :L: (e"T is a geometric series with r = e"', so the series converges {::} lrl < 1 {::} lex! < 1 {::}
n =O n=O
1
- 1 < e"' < 1 0 < e"' < 1 {::} x < 0. In that case, the sum of the series is _a_ = - -.
1 - r l -ex
65. After defining J, We use convert (f, parfrac); in Maple, Apart in Mathematica, or Expand Rational and
. . . 0 . fi d th h I . 3n2 + 3n + 1 1 1
s~mpl~fy 10 enve to n at t e genera term IS (n 2 + n)3 = n 3
( )3 . So the nth partial sum is
. n+1
n(1 1 ) ( 1) (1 1) (1 1 ) 1
Sn = ~.:'f1
k3 - (k + 1)3 =
1 - 23 + 23 - 33 + " . + n3 - (n + 1)3 = . 1 - (n + 1)3
The series converges to lim s,. = 1. This can be confirmed by directly computing the sum using
n-oo
s um (f , n=l.. i nfinity) ; (in Maple), Sum [f, {n , 1 , Inf inity}] (in Mathematica), or Calculus Sum
(from 1 to oo) and Simplify (in Derive).
67. For n = 1, a1 = 0 since s1 = 0. For n > 1,
. n- 1 (n - 1) - 1 (n - l)n- (n + l)(n- 2) 2
an = Sn - Sn- 1 = _n _+ _l - (n- 1) + 1 = (n + l)n = -n '(n- +:-1=-)
I ~ I' 1' 1 - 1/n 1
A so, nL;:l a,.,. = n~~ Sn = n~ 1 + 1/n = .
69. (a) The quantity of the drug in the body after the first tablet is 150 m g. After the second tablet, there is 150 mg plus 5%
of the first 150-mg tablet, that is, (150 + 150(0.05)] mg. After the third tablet, the quantity is
(150 + 150(0.05) + 150(0.05) 2 ] = 157.875 mg. After n tablets, the quantity (in mg) is
= W 1 - 0.05 ..
n 1 . . . 150(1 - 0.05") 3000 ( ")
150 + 150 ( 0.05) + ... + 150 ( 0.05 ) - .·We can use Formula 3 to wnte this as
1 _ 0.0 5
_(b) The number of milligrams remaining in the body in the long run is lim ( 3 ~go (1 - 0.05")] = 3 ~go (1- 0) ~ 157.895,
n.-.oo .
only 0.02 mg more than the amount after 3 tablets.
71. (a) The first step in the chain occurs when the local government spends D dollars. The people who receive it spend a
fraction e of those D dollars, that is, D e dollars. Those who receive the De dollars spend a fraction c of it, that is,
D c 2 dollars. Continuing in U1is way, we see U1at the total spending after n transactions is
D(1 en)
Sn = D + De + De 2 + · · · + Dc" - 1 = - by (3).
1 -c
(b) lim Sn = lim D( 1 - c" ) = __..!2_ lim (1 - c") = __..!2_ [since 0 < c < 1 => lim c" = o]
n.-oo n.-+oo 1 - C 1 - C n -oo 1 - C n--.oo .
= D [since c + s = 1] = kD [since k = 1/ s]
8
If c = 0.8, then s = 1 - c = 0.2 and the multiplier is k = 1/ s = 5.
73. f (1 + c)- n is a geometric series with a ,;, (1 + c)- 2 and r = (1 + c)-t, so the series converges when
n = 2
IC1+ c)- 1 1< 1 {::} l1 +el > 1 {::} 1 +e>1or1+e< - 1 {::} c > Oore < 7 2.Wecalculatethesumofthe
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