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

86 0 CHAPTER 11 INFINITE SEQUENCES AND SERIES
25. {/1 -x = [1 +(-x)FI 4 = f ( 1 / 4 ) ·(-x)"= 1+t(-x)+ H-, ~ )( -x) 2 + H-~),(-D( -x? + · ..
n=O n 2 3.
= 1
_ !x+ ~ (- 1)"- 1 (-1)" · [3 · 7 · · · · · (4n - 5)]x"
4 n=2 4n · n!
1
00
3 · 7 .. · .. (4n - 5)
= 1- - x- I: xn
4 n=2 4n. n !
and 1-xl < 1 ¢} lxl < 1, so R = 1.
27.
1
1
3
=
(2 + x) [2(1 + x/2)] n=O n
(
3
= - 8
1 ( 1 + ~ 2 ) - 3 = - 8
1
f: (- 3 ) (~ 2
) ". The binomial coefficient is
- 3) = (-3)(-4)(- 5) ..... (-3 - n + 1) = ·c-3)(- 4)( -5) ..... [- (n + 2)]
n n! n!
_ ( - 1)" · 2 · 3 · 4 · 5 · · .. · (n + l )(n + 2) _ ( - l)"(n + l )(n + 2)
- 2 ·n! - 2
I
Thus 1 = ! ~ ( -l)"(n + 1)(n + 2) x" = ~ ( -1)"(n + 1)(n + 2)x" for ~ ~~ < 1 ¢} lxl < 2, so R = 2.
' (2 + x) 3 8 n=O 2 2n n=O 2"+ 4 2 .
00 X2n+ l 00 ( 1fX) 2n+ 1 oo 7r2n+ 1
29. sin x= L::(-1)" ( 2 1
) 1
~ f(x) =sin(1rx) = L::(- 1)" ( 2 1
) 1
= L::(-1)" ( 2 1
) 1
x 2 ~+ 1 ,R = oo.
n=O n + · n = O n + · n =O n + ·
oo x"
31. ex = E I
n =O n.
R =oo.
2x ~ (2x)" ~ 2" x" J( ) x + 2o: ~ 1 n ~ 2" n ~ 2" + 1 n
=> e = L.J -
n=O n. n=O n n=O n. n =O n. n = O n.
1 - = L.J - 1 - ,so x =e e = L.J 1 x + L.J 1 x = L.J - - 1 -x,
00 x2n
33. cosx = I: (-1)" (2 ·),
n = O n ·
35. We must write the binomial in the form (1 +expression), so we' ll factor out a 4.
X X :i; X ( • X2)-l/ 2 X oo (-1)
(X2)"
J4 + x 2 = y'4(1 + x 2 /4) = 2 y'1 + x2 /4 = 2 1 + 4 = 2 ,~0 n 2 4
X X ~ ( l )"' 1 · 3 · 5 · · · · · (2n- 1) 2n
= - + - ~ - X
2 2 n=l 2" . 4" . n!
_
X
00
n 1 · 3 · 5 · · · · · (2n- 1) 2n+l X 2 !xj
- 2 + .. ~ 1 {-1) n! 23 ,.+ 1 x and 4 < 1 ¢> 2
< 1 ¢> lxl