Thomas Calculus 13th [Solutions]

Section 10.10 The Binomial Series and Applications of Taylor Series 781
41.
42.
1 1 1
2! 3! 4!
1 1 e e
1
1
3
1
4
1
5
1
3
1 1
2
1
1 1 1 4 1
4 4 4 4 4 4 64 1 1/4 64 3 48
43.
2 4 6
2 4 6
1 3 3 3 1 1 3 1 3 1 3 cos 3
2 4 6
4 2! 4 4! 4 6!
2! 4 4! 4 6! 4 4
1 1 1 1 1 1 1
2
1 1
3
1 1
4
ln 1 1 ln 3
2 2 2 3 2 4 2 2 2 2 3 2 4 2 2 2
44.
2 3 4
45.
46.
47.
3 5 7
3 5 7
1 1 1
3 3 5 7 3 3! 3 5! 3 7! 3 3 2
3 3! 3 5! 3 7!
3 5 7
3 5 7
2 2 2 2 2 1 2 1 2 1 2 1
tan 2
3 3 5 7
3 3 3 5 3 7 3 3 3 5 3 7 3 3
3 4 5 6 3 2 3 3
1 1 x
1 x 1 x
x x x x x x x x x
3
sin
3
48.
2 2 4 4 6 6
3 3 3 1 2 1 4 1 6
1 x x x 1 (3 x) (3 x) (3 x) cos(3 x)
2! 4! 6! 2! 4! 6!
49.
3 5 7 9 3 2 2
2
2
3
3
3
1 1 x
1 x 1 x
x x x x x x x x x
2 2
50.
2 3 4
2 3
2 4 2 5 2 6
2 2 2
2 (2 ) (2 ) (2 )
2 2
2 x x x
x x x x
1 2
2! 3! 4! 2! 3! 4!
x x x x x e
51. 1 2x 3x 2 4x 3 5x 4 d 1 x x 2 x 3 x 4 x
5 d 1 1
dx dx 1 x 2
(1 x)
52.
2 3 4 2 3 4 5
1 1
ln(1 )
1 x x x x x
x x x x x
ln(1 x)
2 3 4 5 x 2 3 4 5
x x
53.
2 3 4 2 3 4 3 5
ln 1 x 1 ln(1 x ) ln(1 x ) x x x x x x x x 2 3 4 2 3 4 2 x x x
x
3 5
54.
n 1 n n 1 n
2 3 4 ( 1) x
( 1) x
ln(1 x) x x x x
error
1 when x 0.1;
2 3 4 n n n
n10
1 1
n
n10 10
8
n 8
n 10 10 when n 8 7 terms
55.
n 1 2 n 1 n 1 2 n 1
1
3 5 7 9 ( 1) x
( 1) x
tan x x x x x x
error
1 when x 1;
3 5 7 9 2n 1 2n 1 2n
1
1 1
10
2n
1 3 2
n 1001 500.5 the first term not used is the
st
501 we must use 500 terms
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