Thomas Calculus 13th [Solutions]

Section 10.10 The Binomial Series and Applications of Taylor Series 777
10.10 THE BINOMIAL SERIES AND APPLICATIONS OF TAYLOR SERIES
1.
1 1
2 2
2 1 1 3
2 2 2
1/2
x
x
1 1 1 2 1 3
(1 x) 1 x 1 x x x
2 2! 3! 2 8 16
3
2.
3.
1 2 2 1 2 5 3
x
x
3 3 3 3 3
1/3 1 1 1 2 5 3
(1 x) 1 x 1 x x x
3 2! 3! 3 9 81
3 ( 3)( 4) 2 ( 3)( 4)( 5) 3 2 3
(1 x) 1 ( 3)( x) ( x) ( x) 1 3x 6x 10x
2! 3!
4.
1 1
2 2
2 1 1 3
2 2 2
1/2
( 2 x)
( 2 x)
1 1 2 1 3
(1 2 x) 1 ( 2 x) 1 x x x
2 2! 3! 2 2
3
5.
x
2
x
3
2 2
2
( 2)( 3) ( 2)( 3)( 4)
3 2 1 3
1 x 1 2 x 1 x x x
2 2 2! 3! 4 2
6.
x
2
x
3
x
4
3 3 3
4
(4)(3) (4)(3)(2) (4)(3)(2)(1)
4 2 2 4 3 1 4
1 x 1 4 x 0 1 x x x x
3 3 2! 3! 4! 3 3 27 81
7.
3
2
3
3
1 3
x
1 3 5
x
2 2 2 2 2
3
1/2
1 3 1 3 3 6 5 9
1 x 1 x 1 x x x
2 2! 3! 2 8 16
8.
2
2
2
3
1 4
x
1 4 7
x
3 3 3 3 3
2
1/3
1 2 1 2 2 4 14 6
1 x 1 x 1 x x x
3 2! 3! 3 9 81
9.
1 1 1
2
1 1 3 1
3
2 2 x 2 2 2 x
1/2
1 1 1 1 1 1 1 1 1
x 2 x 2! 3! 2x 8x 16x
2 3
10.
3
x
1 x
1 4 2 1 4 7 3
x
x
3 3 3 3 3
1/3 1 1 2 2 3 14 4
x(1 x) x 1 x x x x x
3 2! 3! 3 9 81
11.
12.
2 3 4
4 (4)(3) x (4)(3)(2) x (4)(3)(2) x
2 3 4
(1 x) 1 4x 1 4x 6x 4x x
2! 3! 4!
2
2
2
3
3
(3)(2) x (3)(2)(1) x
2 2 2 4 6
1 x 1 3x 1 3x 3x x
2! 3!
13.
14.
2 3
3 (3)(2)( 2 x) (3)(2)(1)( 2 x)
2 3
(1 2 x) 1 3( 2 x) 1 6x 12x 8x
2! 3!
x
2
x
3
x
4
2 2 2
4
(4)(3) (4)(3)(2) (4)(3)(2)(1)
3 2 1 3 1 4
1 x 1 4 x 1 2x x x x
2 2 2! 3! 4! 2 2 16
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