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

SECTION 11.9 REPRESENTATIONS OF FUNCTIONS AS POWER SERIES D 79
3
11. f (x) =
2 2 x - x-
3 A B
..,....------,-,-,---"7 = --+ -- => 3 = A(x + 1) + B(x - 2). Let x = 2 to get A = 1 and
(x- 2)(x + 1) x - 2 x + 1 ·
x = - 1 to get B = - 1. Thus
3 __ 1
x2- x - 2 - X- 2 X+ 1 - 2 1- (x/2) 1 - (- x) 2 n=O 2 n=O
_ _ _ 1_=~( 1 )- 1 =-~ E (=)" _ E(-x)n
= f: [-~ (1:)" -1(-1)"']xn = E [(- 1t+l- n~l]x"
n=O 2 2 n=O 2
We represented f as the sum of two geometric series; the first converges for x E (-2, 2) and the second converges for ( -1, ~) .
Thus, the sum converges for x E ( - 1, 1) =I.
13. (a) f(x ) = 1 = ·-d (-- -1 ) = , --d [ L..... ~ (- 1)n x "]
(1+x) 2 dx 1+x dx n=O
[from Exercise 3]
~ ~
= L (-1)"+ 1 nx"- 1 [from Theorem 2(i)] = L (-1)"(n + l)x" with R = ~ .
n=l
n=O
In the last step, note that we decreased the initial value of the summation variable n by 1, and then increased each
occurrence ofn in the term by 1 [also note that (-1)"+ 2 = (- 1)" ].
(b)f(x)=
1
1
=-1:j_[ ] =-1:j_[E(-1)"(n+ 1)x"] [frompart(a)]
(1+x) 3 2dx (1+x) 2 2dx n=O
~ ~
= -~ L (-1)"(n + 1)nxn- 1 = ~ L (- 1)n(n + 2)(n + 1)x" with R = 1.
n=l
n=O
x 2 1 1 ~
(c)f(x) = ( 1
+x) 3 =x 2 · (l+x) 3 = x 2 · 2nJ;
0
{- 1)''(n+2)(n+1)x" [frompart(b)]
= 1: E (-1)"{n+2)(n+ l)x"+~
2 n=O
To write the power series with x " rather ~an ·x"+ 2 , we will decrease each occurrence of n in the term by 2 and increase
1 . CX>
the initial value of the summation variable by 2. This gives us 2 .. (-1)"(n)(n- 1)x,. with R = 1.
~2
15. f(x) = ln(5 - x) = -~~ = _1: ~~ = -1:1 [f: (::\"] dx = C- ~ E xn+l = ·C..:... f: x"
5-x 5 1 - x/5 5 n=O 5} 5n=o5"(n+ 1) n=tn5n
Putting x = 0, we get C = ln 5. The series converges for Jx/51 < 1 Jx J < 5, so R = 5.
1 1
00
17. We know that - 1 4
= ( 4
) = L ( -4x)". Differentiating, we get
+ X 1- - X n=O
f(x) = {1+x4x)2 = · -4x.
for J-4xJ < 1 Jxl < ;}, so R = ;} .
-4 2 =-X f;{- 4)n+1 (n+1)xn = f;(-1)n4n(n+1)xn+l
(1 + 4x) 4 n=O n=O
® 2012 Ccngage lc.aming. All Rights Rescrvct.l. Moy not be scanned, copic.-d. or duplicntcd. or posted to u pllblic!y ucccssiblc: website. in whole: or in pan.