Thomas Calculus 13th [Solutions]

Section 10.8 Taylor and Maclaurin Series 767
29.
x x x ( n)
x
f ( x) e f ( x) e , f ( x) e f ( x) e ;
2 2 ( n) 2
f (2) e , f (2) e , f (2) e
x
2 3 2
2 2 e 2 e 3
e
2 3! n!
n 0
e e e ( x 2) ( x 2) ( x 2) ( x 2)
n
30.
x x x 2 x 3 ( n)
x n
f ( x) 2 f ( x) 2 ln 2, f ( x) 2 (ln 2) , f ( x) 2 (ln 2) f ( x) 2 (ln 2) ;
2 3 ( n)
f (1) 2, f (1) 2ln 2, f (1) 2(ln 2) , f (1) 2(ln 2) , , f (1) 2(ln 2)
n
x
2 3
2(ln 2) 2 2(ln 2) 3
2(ln 2) ( x 1)
2 3! n!
n 0
2 2 (2ln 2)( x 1) ( x 1) ( x 1)
n
n
31.
32.
f ( x) cos 2 x , f ( x) 2sin 2 x , f ( x) 4cos 2 x , f ( x) 8sin 2 x ,
2 2 2 2
(4) 4 (5) 5
f ( x) 2 cos 2 x , f ( x) 2 sin 2 x , ..;
2 2
(4) 4 (5) (2 n) n 2n
1, 0, 4, 0, 2 , 0, , ( 1) 2
4 4 4 4 4 4 4
f f f f f f f
2 4 n 2n 2n
2
( 1) 2
2 4 3 4
n 0
(2 n)! 4
cos 2 x 1 2 x x x
1 1/2 1 3/2 3 5/2 (4) 15 7/2
2 4 8 16
1 1 3 (4) 15 1 1 2 1 3 5 4
2 4 8 16 2 8 16 128
f ( x) x 1, f ( x) ( x 1) , f ( x) ( x 1) , f ( x) ( x 1) , f ( x) ( x 1) , ;
f (0) 1, f (0) , f (0) , f (0) , f (0) , x 1 1 x x x x
33. The Maclaurin series generated by cos x is
n
n
( 1) 2n
(2 n)!
0
x which converges on ( , ) and the Maclaurin
series generated by
2
2
1 x
2
1 x is n 0
f ( x) cos x is given by
n
x which converges on ( 1, 1). Thus the Maclaurin series generated by
n
( 1) 2n
n
5 2
(2 n)! 2
n 0 n 0
x 2 x 1 2 x x . which converges on the
intersection of ( , ) and ( 1, 1), so the interval of convergence is ( 1, 1).
34. The Maclaurin series generated by
x
e is
n
n
x
n!
0
which converges on ( , ). The Maclaurin series generated
by
2
f ( x) 1 x x e is given by
x
n
2 x 1 2 2 3
n! 2 3
n 0
1 x x 1 x x . which converges on ( , ).
35. The Maclaurin series generated by sin x is
series generated by ln(1 x ) is
( 1)
n
n 1
by f x sin x ln(1 x ) is given by
n 1
( 1) 2n
1
(2n
1)!
n 0
n
n
x which converges on ( , ) and the Maclaurin
x which converges on ( 1, 1). Thus the Maclaurin series generated
n
n 1
( 1) 2n
1 ( 1) n 2 1 3 1 4
(2n
1)! n
2 6
n 0 n 1
x x x x x . which
converges on the intersection of ( , ) and ( 1,1), so the interval convergence is ( 1, 1).
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