Thomas Calculus 13th [Solutions]

798 Chapter 10 Infinite Sequences and Series
22. A polynomial has only a finite number of nonzero terms in its Taylor series, but the functions sin x, ln
x
e have infinitely many nonzero terms in their Taylor expansions.
x and
23.
3 3 3
ax a x x
x
x
sin( ax) sin x x 3! 3!
3 5
a 2 a 1 a 1 2
3 3 2
lim lim lim
x 0 x x 0 x x 0 x
3
if a 2 0 a 2; lim sin 2x sin x x 2 1 7
3
x 0 x
3! 3! 6
3! 3! 5! 5!
x is finite
24.
a
2
x
2
a
4
x
4
b
2 4!
1
cos ax b 2 2 2
lim 1 lim 1 lim 1 b a a x 1 b 1 and a 2
2 2 2
x 0 2x x 0 2x x 0 2x
4 48
25. (a)
u
u
n
n 1
2
2 2
( n 1)
1 2 1 C 2 1 and 1
n n
2
n
n 1 n
un
(b) n 1 1 1 0 C 1 1 and 1
u 2
n 1 n n n
n 1 n diverges
converges
26.
5
2
6 3
n
2
4 2 4n
2
4n
1
1 1
2 2 2 2
un
2 n(2n 1) 4n 2n
5
un
1 (2n 1) 4n 4n 1 n 4n 4n 1 n n
after long division C 3 1 and
2
2
f ( n) 5n
5 5 u
2
n converges by Raabes Test
4n
4n
1 4 n 1
4 1
n
n
2
27. (a)
2 2
an L an an an anL a n converges by the Direct Comparison Test
n 1 n 1 n 1
(b) converges by the Limit Comparison Test:
lim an
0
n
a n
1 a n
1
lim lim 1 since
n
an
n
1 an
n
a n converges and therefore
1
2 3
an an 2 3 an
28. If 0 a n 1 then ln 1 an ln 1 an an a ,
2 3 n an a n a positive term of a
1 an
convergent series, by the Limit Comparison Test and Exercise 27b
29.
1
n
(1 x) 1 x where 1
1 n 1
x 1 d (1 x)
nx and when x 1 we have
2
(1 x)
dx
2
n 1
n 1
1 1
2
1
3
1
n 1
4 1 2 3 4 n
2 2 2 2
30. (a)
2 2
n 1 n 2 n 1 2
n
x x ( n 1) x x x n( n 1) x n( n 1) x 2x
1 x (1 x) (1 x) (1 x)
n 1 n 1 n 1 n 1
2 3 3
n
1
2
x
2
x
3
1 ( x
3
1)
x
n( n 1) 2
x
n
1
, x 1
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