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

104 0 CHAPTER 11 INFINITE SEQUENCES AND SERIES
(b) 1.5
(c) IR3 (x) l $ ~~ ix - 11 4 , where J! ( 4 ) (x) J $ M with
j ( 4 >(x) = - ~~x- 7 1 2 . Now 0.9 ~ x $ 1.1 =>
- 0.1 ~ X- 1 ~ 0.1 => (x- 1) 4 $ ·(0.1) 4 ,
d I . 0 . . 15
an etting x = .9 gtves M = ( _ )7 ,
16 0 9 12
so
0
2 IRa(x) l $
16 (0. ~~ 712 41 (0.1)4 :::::: 0.000005648
:::::: 0.000 006 = 6 X lO- G
(d) 5 X 10- 6 From the graph of IR3(x)l = I JX - T3(x)l. it appears that
the error is less than 5 x w- 6 on [0.9, 1.1].
oo • x2n+l x3 x6 x7 . x3 xs x7
59.sin x = 2:(- 1t ( 2 1
) 1
= x- - +---+· ··,sosmx - x=--+---+··· and
n=O n + . 3! 5! 7! 3! 5! 7!
00 00 00
61. f(x ) = 2: Cn x" => f(-x)= l: cn(-xt = 2:(- 1)"c.,x"
n=O n =O n=O
(a) Iff is an odd function, then f( - x) = - f(x) =>
00 00
2: (- 1)"cnx" = 2: - c,,x". The coefficients of any power series
u=O
are uniquely determined (by Theorem 11.10.5), so ( -1)" Cn = - en.
If n is even, then ( - 1)" = 1, so en = -c,. => 2en = 0 => c,. = 0. Thus, all even coefficients are 0, that is,
Co = C2 = C4 = · · · = 0.
00 00
(b) Iff is even, then f (-x) = f(x) => 2: {-lt c,. xn = 2: Cn x" => {-1)" Cn = Cn.
n=O
n = O
If n is odd, then ( - 1 )" = -1, so -en = Cn => 2c,.. = 0 => Cn = 0. Thus, all odd coefficients are 0,
that is, Cl = C3 = C5 = · · · = 0.
n=O
'
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