638 Solutions to exercises
- 11 − 13 x 1 + 13 x
2
1 − 1 x
3
- 11 + 112 x 1 + 154 x
2
1 + 1108 x
3
1 + 181 x
4
- 11 − 120 x 1 + 1160 x
2
1 − 1640 x
3
1 + 11280 x
4
1 − 11024 x
5
- 811 − 1216 x 1 + 1216 x
2
1 − 196 x
3
1 + 116 x
4
- 7291 + 11458 x 1 + 11215 x
2
1 + 1540 x
3
1 + 1135 x
4
1
- 118 x
5
1 + 1 x
6
- (i)
(ii)(a
4
1 + 1 b
4
1 + 1 c
4
) 1 + 1 4(a
3
b 1 + 1 a
3
c 1 + 1 b
3
a 1 + 1 b
3
c 1
- 1 c
3
a 1 + 1 c
3
b) 1 + 1 6(a
2
b
2
1 + 1 a
2
c
2
1 + 1 b
2
c
2
) 1
- 1 12(a
2
bc 1 + 1 b
2
ca 1 + 1 c
2
ab)
- (i)
(ii)(a
3
1 + 1 b
3
1 + 1 c
3
1 + 1 d
3
) 1 + 1 3(a
2
b 1 + 1 a
2
c 1 + 1 a
2
d 1
- 1 b
2
a 1 + 1 b
2
c 1 + 1 b
2
d 1 + 1 c
2
a 1 + 1 c
2
b 1 + 1 c
2
d 1
- 1 d
2
a 1 + 1 d
2
b 1 + 1 d
2
c) 1 + 1 6(abc 1 + 1 abd 1
- 1 acd 1 + 1 bcd)
10211 37.
Section 7.4
- (i) 11 + 13 x 1 + 19 x
2
1 + 127 x
3
1 + 181 x
4
1 + 1243 x
5
1
- 1729 x
6
1 +1-
(ii)|x| 1 < 13
- (i) 11 − 15 x
2
1 + 125 x
4
1 − 1125 x
6
1 +1-
(ii)
- (i)
(ii)|x| 1 < 12
(i)0.000001000001000001 -
Section 7.5
46.diverges 47.converges
48.converges for all a
49.no conclusion (see Exercise 50)
50.diverges ifa 1 ≤ 11 , converges ifa 1 > 11
51.diverges
qe
T
v
v
=−
−
−
1
1
θ
1
2 4 8 16 32 64 128
2345 6
−+ − + − + −
xx x x x x
||x< 15
n
nnn
2
432
12
2651 ++−
3
4
23
21 2
−
++
n
()( )nn
3
1110
6
!
!!!!
=
3
3000
1
3
2100
3
!
!!!!
,
!
!!!!
==,
4
211
12
!
!!!
=
4
400
1
4
310
4
4
220
6
!
!!!
,
!
!!!
,
!
!!!
== =,
Section 7.6
4 53. 1 54. 1
1 56. 0 57.
11 − 1 x
2
1 + 1 x
4
1 − 1 x
6
1 + 1 x
8
1
- (i)
(ii)
- (i) (ii) 01 < 1 x 1 < 12
- (i) (ii)all x
- (i) (ii)all x
- (i) (ii) 01 < 1 x 1 ≤ 14
Section 7.7
- (i)
(ii)2.08008214
(iii)
- 1 74. 122 75.− 2 76. 122
2 0800832 9 2 0800840
3
.<<.
2
12 288
5
20736
5
248832
23 4
+− + −
xx x x
ln 2
12
2
1
−
−
=
∑
n
n
n
x
∞
n
nn
x
n
=
∑
−−
0
2
1
2
∞
()( )
()!
π 2
e
x
n
n
n
2
0
2
=
∑
−
!
∞
()
n
n
x
=
∑
−
0
1
∞
()
Bb
ab
RT
a
RT
2
2
2
22
2
=− +
BBb
a
RT
01
==−, 1 ,
BBb
a
RT
Bb
01 2
2
==−,= 1 ,
→→
1
2
0
0
2
mcvv as
Tm
cc
=+
1
2
1
3
4
5
8
0
2
24
v
vv
x
xxx x
++++
357 9
2 6 24 120
13
9
2
9
2
27
8
23 4
−+ − +x
xx x
−− − 2 − −
8
3
4
32
5
32
3
2
34
x
x
xx
2
4
3
4
15
8
315
4
2835
2
6101418
x
xx x x
−+ − +
1
3 9 27 81 243
23 4
−+ − +
xx x x
1
2
3
8
5
16
35
128
23 4
++ + +
xx x x
1
39
5
81
10
243
23 4
+− + −
xx x x
3