Boas Solutions

Chapter 1

1.1
1.3
1.6
1.9
1.13
1.16

(2/3)10 = 0.0173 yd; 6(2/3)10 = 0.104 yd (compared to a total of 5 yd)
5/9
1.4 9/11
1.5 7/12
11/18
1.7 5/27
1.8 25/36
6/7
1.10 15/26
1.11 19/28
$1646.99
1.15 Blank area = 1
At x = 1: 1/(1 + r); at x = 0: r/(1 + r); maximum escape at x = 0 is 1/2.

2.1
2.4
2.7

1
∞
e2

4.1
4.2
4.3
4.4
4.5

= 1/2n → 0; Sn = 1 − 1/2n → 1; Rn= 1/2n → 0
= 1/5n−1 → 0; Sn = (5/4)(1 − 1/5n ) → 5/4; Rn = 1/(4 · 5n−1 ) → 0
= (−1/2)n−1 → 0; Sn = (2/3)[1 − (−1/2)n ] → 2/3; Rn = (2/3)(−1/2)n → 0
= 1/3n → 0; Sn = (1/2)(1 − 1/3n ) → 1/2; Rn = 1/(2 · 3n ) → 0
= (3/4)n−1 → 0; Sn = 4[1 − (3/4)n ] → 4; Rn = 4(3/4)n → 0
1
1
1
→ 0; Sn = 1 −
→ 1; Rn =
→0
an =
n(n + 1)
n+1
n+1
1
(−1)n+1
1
(−1)n
an = (−1)n+1
→ 0 ; Sn = 1 +
+
→ 1; Rn=
→0
n n+1
n+1
n+1

4.6
4.7

2.2
2.5
2.8

1/2
0
0

2.3
2.6
2.9

0
∞
1

an
an
an
an
an

5.1
5.4
5.7
5.9

D
D
Test further
D

5.2
5.5
5.8
5.10

6.5
Note:
6.7
6.10
6.13
6.19
6.22
6.25
6.28
6.32
6.35

(a) D
6.5
In the following answers,
D, I = ∞
6.8
C, I = π/6
6.11
D, I = ∞
6.14
C, ρ = 3/4
6.20
C, ρ = 0
6.23
C, ρ = 0
6.26
C, ρ =4/27
6.29
D, cf.
n −1
6.33
C, cf.
n −2
6.36

Test further
D
Test further
D

5.3
5.6

Test further
Test further

(b) D
∞
I=
an dn; ρ = test ratio.
D, I = ∞
6.9 C, I = 0
C, I = 0
6.12 C, I = 0
D, I = ∞
6.18 D, ρ = 2
C, ρ = 0
6.21 D, ρ = 5/4
D, ρ = ∞
6.24 D, ρ = 9/8
C, ρ = (e/3)3
6.27 D, ρ = 100
D, ρ = 2
6.31 D, cf.
n −1
−n
C, cf.
2
6.34 C, cf.
n −2
−1/2D, cf.
n

1

Chapter 1

2

7.1
7.5

C
C

7.2
7.6

D
D

9.1
9.3
9.5
9.7
9.9
9.11
9.13
9.15
9.17
9.19
9.21
9.22

D, cf.
n −1
C, I = 0
C, cf.
n −2
D, ρ = 4/3
D, ρ = e
D, I = ∞, or cf.
n −1
C, I = 0, or cf.
n −2
D, ρ = ∞, an → 0
C, ρ = 1/27
C
C, ρ = 1/2
(a) C
(b) D

10.1
10.4
10.7
10.10
10.13
10.16
10.19
10.22
10.25

|x| < √
1
10.2
|x|≤ 2
10.5
−1 ≤ x < 1
10.8
|x| ≤ 1
10.11
−1 < x ≤ 1
10.14
−1 < x < 3
10.17
|x| < 3
10.20
No x
10.23
nπ − π/6 < x < nπ + π/6

7.3
7.7

C
C

7.4
7.8

9.2
9.4
9.6
9.8
9.10
9.12
9.14
9.16
9.18
9.20

D, an → 0
D, I = ∞, or cf.
C, ρ = 1/4
C, ρ = 1/5
D, an → 0
C, cf.
n −2
C, alt. ser.
C, cf.
n −2
C, alt. ser.
C

C
C
n −1

(c) k > e
|x| < 3/2
All x
−1 −5 ≤ x < 5
|x| < 3
−2 < x ≤ 0
All x
x > 2 or x < −4

10.3
10.6
10.9
10.12
10.15
10.18
10.21
10.24

−1/2
(−1)n (2n − 1)!!
−1/2
= 1;
=
(2n)!!
0
n
Answers to part (b), Problems 5 to 19:
∞ n+2
∞
x
13.5 −
13.6
n
1
0

|x| ≤ 1
All x
|x| < 1
|x| < 1/2
−1 < x < 5
−3/4 ≤ x ≤ −1/4
0 ≤ x√ 1
≤
|x| < 5/2

13.4

13.7

∞
0

(−1)n x2n
(2n + 1)!

13.9 1 +2
13.11
13.13
13.15
13.16
13.18
13.20
13.21
13.22
13.23

∞
0
∞
0
∞
0
∞
0
∞

∞

13.8

xn

13.10

1

(−1)n xn
(2n + 1)!

13.12

(−1)n x2n+1
n!(2n + 1)

13.14

1/2 n+1
x
(see Example 2)
n

∞

0
∞
0
∞
0
∞
0

−1/2
(−x2 )n (see Problem 13.4)
n
(−1)n x4n+2
(2n + 1)!
(−1)n x4n+1
(2n)!(4n + 1)
x2n+1
2n + 1

2n+1

x
−1/2
(−1)n
n
2n + 1x2n
(2n)!

∞

13.17 2

(−1)n x2n+1
13.19
(2n + 1)(2n + 1)!
0
x + x2 + x3 /3 − x5 /30 − x6 /90 · · ·
x2 + 2x4 /3 + 17x6 /45 · · ·
1 + 2x + 5x2 /2 + 8x3 /3 + 65x4 /24 · · ·
1 − x + x3 − x4 + x6 · · ·

o ddn
∞
0

xn
n

−1/2 x2n+1
n
2n + 1

Chapter 1
13.24
13.25
13.26
13.27
13.28
13.29
13.30
13.31
13.32
13.33
13.34
13.35
13.36
13.37
13.38
13.39
13.4013.41
13.42
13.43
13.44

1 + x2 /2! + 5x4 /4! + 61x6 /6! · · ·
1 − x + x2 /3 − x4 /45 · · ·
1 + x2 /4 + 7x4 /96 + 139x6 /5760 · · ·
1 + x + x2 /2 − x4 /8 − x5 /15 · · ·
x − x2 /2 + x3 /6 − x5 /12 · · ·
1 + x/2 − 3x2 /8 + 17x3 /48 · · ·
1 − x + x2 /2 − x3 /2 + 3x4 /8 − 3x5 /8 · · ·
1 − x2 /2 − x3 /2 − x4 /4 − x5 /24 · · ·
x + x2 /2 − x3 /6 − x4 /12 · · ·
1 + x3 /6 + x4 /6 + 19x5 /120...

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