APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES
||||
A101
41.
43.
All curves have the vertical asymptote . For , the
curve bulges to the right. At , the curve is the line .
For , it bulges to the left. At there is a cusp at
(0, 0). For , there is a loop.
45. 47.
49. 51.
53. 55.
57.
,
PROBLEMS PLUS
N
PAGE 708
1.
3.
5.
(a) At (0, 0) and
(b) Horizontal tangents at (0, 0) and ;
vertical tangents at (0, 0) and
(d)
(g)
CHAPTER 12
EXERCISES 12.1
N
PAGE 720
Abbreviations: C, convergent; D, divergent
1. (a) A sequence is an ordered list of numbers. It can also be
defined as a function whose domain is the set of positive integers.
(b) The terms approach 8 as becomes large.
(c) The terms become large as becomes large.
3. 0.8, 0.96, 0.992, 0.9984, 0.99968 5.
7.
3, 5, 9, 17, 33 9. 11.
13. 15.
; yes;
17. 1 19. 5 21. 1 23. 1 25. 0 27. D
29. 0 31. 0 33. 0 35. 0 37. 1 39.
41. 43.
D
45.
D
47.
1
49.
51.
D
53.
0
55. (a) 1060, 1123.60, 1191.02, 1262.48, 1338.23 (b) D
1
2
ln 2
e
2
1
2
1
3
,
2
5
,
3
7
,
4
9
,
5
11
,
6
13
a
n
(
⫺
2
3
)
n⫺1
a
n
5n ⫺ 3a
n
1兾共2n ⫺ 1兲
⫺3,
3
2
, ⫺
1
2
,
1
8
, ⫺
1
40
na
n
na
n
3
2
(
⫺
25
24
, 3
)
, 共⫺1, 3兲共⫾1, 0兲, 共⫾3, 0兲
c ⬎ 0
c 0⫺1
⬍
c
⬍
0
x 1c ⫺1
c
⬍
⫺1x 1
471,295
兾1024
57.
59.
Convergent by the Monotonic Sequence Theorem;
61. Decreasing; yes 63. Not monotonic; no
65. Decreasing; yes 67. 2 69.
71.
(b) 73. (a) 0 (b) 9, 11
EXERCISES 12.2
N
PAGE 730
1. (a) A sequence is an ordered list of numbers whereas a series is
the sum of a list of numbers.
(b) A series is convergent if the sequence of partial sums is a con-
vergent sequence. A series is divergent if it is not convergent.
3. ,,
,,
,,
,,
,;
convergent,
5. ,,
,,
,,
,,
,;
divergent
7. ,,
,,
,,
,,
,;
convergent, sum
9. (a) C (b) D 11. 9 13. D 15. 60 17.
19.
D 21. D 23. D 25. 27.
D
29.
D
31. D 33. 35. 37. 39.
41. 43. 45.
47. 49.
51.
All ; 53. 1
55. for ,
57. (a) (b) 5 59.
63. 65.
The series is divergent.
1
n共n ⫹ 1兲
1
2
(
s
3
⫺ 1
)
S
n
D共1 ⫺ c
n
兲
1 ⫺ c
sum 1n ⬎ 1a
1
0, a
n
2
n共n ⫹ 1兲
2
2 ⫺ cos x
x
⫺
1
4
⬍
x
⬍
1
4
;
1
1 ⫺ 4x
⫺3
⬍
x
⬍
3;
x
3 ⫺ x
5063兾33001138兾333
2
9
e ⫺ 1
11
6
3
2
e兾共e ⫺ 1兲
5
2
1
7
1
0.698490.68377
0.666670.64645
0.622040.59175
0.552790.50000