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

SECTION 11.2 SERIES 0 55
47. F~r the series .. ~ 1
( e 1 1" - e 1 / (n+l)) ,
S n = i~ ( el/i _ el/(i+I)) = (el _ el /2) + (el /2 _ el/ 3) + ... + ( el/n _ el / (n+l)) = e _ el /(n+l)
[telescoping series]
Thus, f: (el/n - el/ (n+l)) = lim S n = lim (e - e 1 / (n+l)) = e - e 0 = e- 1.
n=l n-oo n--+oo
Converges
49. (a) Many people would guess that x < 1, but note that x consists of an infinite number of9s.
9 9 9 9 ~ 9 l " h " . . .
(b) x = 0.99999 .. . = + + 10 100 1000
+ , 10 000
+ ··· = .~
1 10 .. , w 11c 1s a geo~etnc senes w1th a 1 = 0.9 and
. 0.9 0.9 1 h . 1
r = 0.1. Its sum IS 1
_ 0. 1
= 0. 9
= , t at 1s, x = .
(c) The number 1 has two decimal representations, 1.00000 . .. and 0.99999 . . . .
(d) Except for 0, all rational nurnbe.rs that have a terminating decimal representation can be written in more than one way. For
example, 0.5 can be written as 0.49999 . . . as well as 0.50000 ... .
- 8 8 . . . .tl 8 d 1 l a 8/10 8
51. 0.8 = 10
+ 102
+ · · · IS a geometriC senes WI 1 a = 10
an r = 10
. t converges to - - = / = -
1 - 1" 1 - 1 10 9 :
- 516 516 "516 516 . . . . 516 1
53. 2.516 = 2 + 103
+ 106
+ · · · . Now 103
+ 106
+ · · · IS a geometnc senes With a= 103
and r = 103
. It converges to
_a_ = 516/ 10 3 = 516/ 10 3 = 516 . Thus. 2 . 516 = 2 516 = 2514 = 838
1 .:... r 1 - 1/ 103 999/ 103 999 ' + 999 999 333 ·
- 42 42 42 42 . . . . 42 1
55. 1.5342 = 1.53 + 104
+ 106
+ · · · . Now 104
+ 106
+ · · · 1s a geometnc senes w1th a = 104
and r = 102
.
a 42/ 10 4 42/ 10 4 42
It converges to 1 - 1· = 1 - 1/ 102 = 99/ 102 = 9900 ·
- 42 153 42 15,147 42 15,189 5063
Thus, 1. 5342 = 1. 53 + 9900 = 100 + 9900 = 9900 + 9900 = 9900 ~r 3300 ·
00 00
57. 2..: (-5)"xn = 2..: (- 5x )" is a geometric series with r = - 5x, so ilie series converges ¢:? lrl < 1 ~
n = l
n=l
5 x
. 1 - r 1 - - 5x = 1 + 5x ·
J- 5xJ < 1 ~ JxJ < t, that is, -t < X< r In that case, the sum of the series is _ a_ = ( 5 X ) -
00
59. 'E
n=O
(x - 2)"
00
(X-
2) " X-
-,
2
so the series converges
3 .. = E - 3 - is a geometric series with r = - 3
n=O
~ Jrl < 1 ~
x - 2
¢:? - 1 < - 3- < 1 - 3 < x- 2 < 3 ~ - 1 < x < 5. In that case, the sum of the series is
a 1
1- r = 1
_ x -2
3
1
3 - (x - 2)
3
3
5- x ·
' ~ 2" ~ (2)". . . . .... 2 h .
61. L..- ---;; = L..- - IS a geometric senes Wlw r = :-• sot e senes converges
n=O X n=O X _- X
~ Jrl < 1
2 < Jxl x > 2 or x < -2. In that case, the sum of the series is - a = __!__
1 2 / = ~ 2 .
1 - r 1 - X X -
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