Chapter 1
1.1 (2/3)10 = 0.0173 yd; 6(2/3)10 = 0.104 yd (compared to a total of 5 yd)
1.3 5/9 1.4 9/11 1.5 7/12
1.6 11/18 1.7 5/27 1.8 25/36
1.9 6/7 1.10 15/26 1.11 19/28
1.13 $1646.99 1.15 Blank area = 1
1.16 At x = 1: 1/(1 + r); at x = 0: r/(1 + r); maximum escape at x = 0 is 1/2.
2.1 1 2.2 1/2 2.3 0
2.4 ∞ 2.5 0 2.6 ∞
2.7 e2 2.8 0 2.9 1
4.1 an = 1/2n → 0; Sn = 1 − 1/2n → 1; Rn = 1/2n → 0
4.2 an = 1/5n−1 → 0; Sn = (5/4)(1 − 1/5n ) → 5/4; Rn = 1/(4 · 5n−1 ) → 0
4.3 an = (−1/2)n−1 → 0; Sn = (2/3)[1 − (−1/2)n ] → 2/3; Rn = (2/3)(−1/2)n → 0
4.4 an = 1/3n → 0; Sn = (1/2)(1 − 1/3n ) → 1/2; Rn = 1/(2 · 3n ) → 0
4.5 an = (3/4)n−1 → 0; Sn = 4[1 − (3/4)n ] → 4; Rn = 4(3/4)n → 0
1 1 1
4.6 an = → 0; Sn = 1 − → 1; Rn = →0
n(n + 1) n+1 n+1
(−1)n+1 (−1)n
1 1
4.7 an = (−1)n+1 + → 0 ; Sn = 1 + → 1; Rn = →0
n n+1 n+1 n+1
5.1 D 5.2 Test further 5.3 Test further
5.4 D 5.5 D 5.6 Test further
5.7 Test further 5.8 Test further
5.9 D 5.10 D
6.5 (a) D 6.5 (b) D
R∞
Note: In the following answers, I= an dn; ρ = test ratio.
6.7 D, I = ∞ 6.8 D, I = ∞ 6.9 C, I = 0
6.10 C, I = π/6 6.11 C, I = 0 6.12 C, I = 0
6.13 D, I = ∞ 6.14 D, I = ∞ 6.18 D, ρ = 2
6.19 C, ρ = 3/4 6.20 C, ρ = 0 6.21 D, ρ = 5/4
6.22 C, ρ = 0 6.23 D, ρ = ∞ 6.24 D, ρ = 9/8
6.25 C, ρ = 0 6.26 C, ρ = (e/3)3 6.27 D, ρ =P100
6.28 C, ρ =P 4/27 6.29 D, ρ =P2 6.31 D, cf. P n−1
6.32 D, cf. n−1 6.33 C, cf. 2−n 6.34 C, cf. n−2
P −2 P −1/2
6.35 C, cf. n 6.36 D, cf. n
1
,Chapter 1 2
7.1 C 7.2 D 7.3 C 7.4 C
7.5 C 7.6 D 7.7 C 7.8 C
P −1
9.1 D, cf. n 9.2 D, an 6→ 0 P −1
9.3 C, I =P0 9.4 D, I = ∞, or cf. n
9.5 C, cf. n−2 9.6 C, ρ = 1/4
9.7 D, ρ = 4/3 9.8 C, ρ = 1/5
9.9 D, ρ = e 9.10 D, an 6→
P 0 −2
D, I = ∞, or cf.P n−1
P
9.11 9.12 C, cf. n
9.13 C, I = 0, or cf. n−2 9.14 C, alt.Pser.
9.15 D, ρ = ∞, an 6→ 0 9.16 C, cf. n−2
9.17 C, ρ = 1/27 9.18 C, alt. ser.
9.19 C 9.20 C
9.21 C, ρ = 1/2
9.22 (a) C (b) D (c) k > e
10.1 |x| < √ 1 10.2 |x| < 3/2 10.3 |x| ≤ 1
10.4 |x| ≤ 2 10.5 All x 10.6 All x
10.7 −1 ≤ x < 1 10.8 −1 < x ≤ 1 10.9 |x| < 1
10.10 |x| ≤ 1 10.11 −5 ≤ x < 5 10.12 |x| < 1/2
10.13 −1 < x ≤ 1 10.14 |x| < 3 10.15 −1 < x < 5
10.16 −1 < x < 3 10.17 −2 < x ≤ 0 10.18 −3/4 ≤ x ≤ −1/4
10.19 |x| < 3 10.20 All x 10.21 0 ≤ x √≤1
10.22 No x 10.23 x > 2 or x < −4 10.24 |x| < 5/2
10.25 nπ − π/6 < x < nπ + π/6
(−1)n (2n − 1)!!
−1/2 −1/2
13.4 = 1; =
0 n (2n)!!
Answers to part (b), Problems 5 to 19:
∞ n+2 ∞
X x X 1/2 n+1
13.5 − 13.6 x (see Example 2)
1
n 0
n
∞ ∞
(−1)n x2n
X X −1/2
13.7 13.8 (−x2 )n (see Problem 13.4)
0
(2n + 1)! 0
n
∞ ∞
X X (−1)n x4n+2
13.9 1 + 2 xn 13.10
1 0
(2n + 1)!
∞ n n ∞
X (−1) x X (−1)n x4n+1
13.11 13.12
0
(2n + 1)! 0
(2n)!(4n + 1)
∞ n 2n+1 ∞
X (−1) x X x2n+1
13.13 13.14
0
n!(2n + 1) 0
2n + 1
∞
x2n+1
X −1/2
13.15 (−1)n
0
n 2n + 1
∞ 2n ∞
X x X xn
13.16 13.17 2
0
(2n)! n
oddn
∞
X (−1)n x2n+1 ∞
X −1/2 x2n+1
13.18 13.19
0
(2n + 1)(2n + 1)! 0
n 2n + 1
2 3 5 6
13.20 x + x + x /3 − x /30 − x /90 · · ·
13.21 x2 + 2x4 /3 + 17x6 /45 · · ·
13.22 1 + 2x + 5x2 /2 + 8x3 /3 + 65x4 /24 · · ·
13.23 1 − x + x3 − x4 + x6 · · ·
,Chapter 1 3
13.24 1 + x2 /2! + 5x4 /4! + 61x6 /6! · · ·
13.25 1 − x + x2 /3 − x4 /45 · · ·
13.26 1 + x2 /4 + 7x4 /96 + 139x6 /5760 · · ·
13.27 1 + x + x2 /2 − x4 /8 − x5 /15 · · ·
13.28 x − x2 /2 + x3 /6 − x5 /12 · · ·
13.29 1 + x/2 − 3x2 /8 + 17x3 /48 · · ·
13.30 1 − x + x2 /2 − x3 /2 + 3x4 /8 − 3x5 /8 · · ·
13.31 1 − x2 /2 − x3 /2 − x4 /4 − x5 /24 · · ·
13.32 x + x2 /2 − x3 /6 − x4 /12 · · ·
13.33 1 + x3 /6 + x4 /6 + 19x5 /120 + 19x6 /120 · · ·
13.34 x − x2 + x3 − 13x4 /12 + 5x5 /4 · · ·
13.35 1 + x2 /3! + 7x4 /(3 · 5!) + 31x6 /(3 · 7!) · · ·
13.36 u2 /2 + u4 /12 + u6 /20 · · ·
13.37 −(x2 /2 + x4 /12 + x6 /45 · · · )
13.38 e(1 − x2 /2 + x4 /6 · · · )
4
13.39 1 − (x − π/2)2 /2! + (x − π/2) /4! · · ·
3
13.40 1 − (x − 1) + (x − 1)2 − (x − 1) · · ·
13.41 e [1 + (x − 3) + (x − 3) /2! + (x − 3)3 /3! · · · ]
3 2
2
13.42 −1 + (x − π) /2! − (x − π)4 /4! · · ·
13.43 −[(x − π/2) + (x − π/2)3 /3 + 2(x − π/2)5 /15 · · · ]
13.44 5 + (x − 25)/10 − (x − 25)2 /103 + (x − 25)3 /(5 · 104 ) · · ·
14.6 Error < (1/2)(0.1)2 ÷ (1 − 0.1) < 0.0056
14.7 Error < (3/8)(1/4)2 ÷ (1 − 14 ) = 1/32
14.8 For x < 0, error < (1/64)(1/2)4 < 0.001
For x > 0, error < 0.001 ÷ (1 − 12 ) = 0.002
1
14.9 Term n + 1 is an+1 = (n+1)(n+2) , so Rn = (n + 2)an+1 .
14.10 S4 = 0.3052, error < 0.0021 (cf. S = 1 − ln 2 = 0.307)
15.1 −x4 /24 − x5 /30 · · · ' −3.376 × 10−16
15.2 x8 /3 − 14x12 /45 · · · ' 1.433 × 10−16
15.3 x5 /15 − 2x7 /45 · · · ' 6.667 × 10−17
15.4 x3 /3 + 5x4 /6 · · · ' 1.430 × 10−11
15.5 0 15.6 12 15.7 10!
15.8 1/2 15.9 −1/6 15.10 −1
15.11 4 15.12 1/3 15.13 −1
15.14 t − t3 /3, error < 10−6 15.15 23 t3/2 − 52 t5/2 , error < 17 10−7
15.16 e2 − 1 15.17 √cos π2 = 0
15.18 ln 2 15.19 2
15.20 (a) 1/8 (b) 5e (c) 9/4
15.21 (a) 0.397117 (b) 0.937548 (c) 1.291286
15.22 (a) π 4 /90 (b) 1.202057 (c) 2.612375
15.23 (a) 1/2 (b) 1/6 (c) 1/3 (d) −1/2
15.24 (a) −π (b) 0 (c) −1
(d) 0 (e) 0 (f) 0
15.27 (a) 1 − vc = 1.3 × 10−5 , or v = 0.999987c
(b) 1 − vc = 5.2 × 10−7
(c) 1 − vc = 2.1 × 10−10
(d) 1 − vc = 1.3 × 10−11
15.28 mc2 + 21 mv 2
15.29 (a) F/W = θ + θ3 /3 · · ·
(b) F/W = x/l + x3 /(2l3 ) + 3x5 /(8l5 ) · · ·
, Chapter 1 4
15.30 (a) T = F (5/x + x/40 − x3 /16000 · · · )
(b) T = 21 (F/θ)(1 + θ2 /6 + 7θ4 /360 · · · )
15.31 (a) finite (b) infinite
16.1 (c) overhang: 2 3 10 100
books needed: 32 228 2.7 × 108 4 × 1086
P −3/2
16.4 C, ρ = 0 16.5 D, an 6→ P 0 −1 16.6 C, cf. n
16.7 D, I = ∞ 16.8 D, cf. n 16.9 −1 ≤ x < 1
16.10 |x| < 4 16.11 |x| ≤ 1 16.12 |x| < 5
16.13 −5 < x ≤ 1
16.14 1 − x2 /2 + x3 /2 − 5x4 /12 · · ·
16.15 −x2 /6 − x4 /180 − x6 /2835 · · ·
16.16 1 − x/2 + 3x2 /8 − 11x3 /48 + 19x4 /128 · · ·
16.17 1 + x2 /2 + x4 /4 + 7x6 /48 · · ·
16.18 x − x3 /3 + x5 /5 − x7 /7 · · ·
16.19 −(x − π) + (x − π)3 /3! − (x − π)5 /5! · · ·
16.20 2 + (x − 8)/12 − (x − 8)2 /(25 · 32 ) + 5(x − 8)3 /(28 · 34 ) · · ·
16.21 e[1 + (x − 1) + (x − 1)2 /2! + (x − 1)3 /3! · · · ]
16.22 arc tan 1 = π/4 16.23 1 − (sinπ)/π = 1
16.24 eln 3 − 1 = 2 16.25 −2
16.26 −1/3 16.27 2/3
16.28 1 16.29 6!
16.30 (b) For N = 130, 10.5821 < ζ(1.1) < 10.5868
16.31 (a) 10430 terms. For N = 200, 100.5755 < ζ(1.01) < 100.5803
16.31 (b) 2.66 × 1086 terms. For N = 15, 1.6905 < S < 1.6952
200 86
16.31 (c) ee = 103.1382×10 terms. For N = 40, 38.4048 < S < 38.4088
1.1 (2/3)10 = 0.0173 yd; 6(2/3)10 = 0.104 yd (compared to a total of 5 yd)
1.3 5/9 1.4 9/11 1.5 7/12
1.6 11/18 1.7 5/27 1.8 25/36
1.9 6/7 1.10 15/26 1.11 19/28
1.13 $1646.99 1.15 Blank area = 1
1.16 At x = 1: 1/(1 + r); at x = 0: r/(1 + r); maximum escape at x = 0 is 1/2.
2.1 1 2.2 1/2 2.3 0
2.4 ∞ 2.5 0 2.6 ∞
2.7 e2 2.8 0 2.9 1
4.1 an = 1/2n → 0; Sn = 1 − 1/2n → 1; Rn = 1/2n → 0
4.2 an = 1/5n−1 → 0; Sn = (5/4)(1 − 1/5n ) → 5/4; Rn = 1/(4 · 5n−1 ) → 0
4.3 an = (−1/2)n−1 → 0; Sn = (2/3)[1 − (−1/2)n ] → 2/3; Rn = (2/3)(−1/2)n → 0
4.4 an = 1/3n → 0; Sn = (1/2)(1 − 1/3n ) → 1/2; Rn = 1/(2 · 3n ) → 0
4.5 an = (3/4)n−1 → 0; Sn = 4[1 − (3/4)n ] → 4; Rn = 4(3/4)n → 0
1 1 1
4.6 an = → 0; Sn = 1 − → 1; Rn = →0
n(n + 1) n+1 n+1
(−1)n+1 (−1)n
1 1
4.7 an = (−1)n+1 + → 0 ; Sn = 1 + → 1; Rn = →0
n n+1 n+1 n+1
5.1 D 5.2 Test further 5.3 Test further
5.4 D 5.5 D 5.6 Test further
5.7 Test further 5.8 Test further
5.9 D 5.10 D
6.5 (a) D 6.5 (b) D
R∞
Note: In the following answers, I= an dn; ρ = test ratio.
6.7 D, I = ∞ 6.8 D, I = ∞ 6.9 C, I = 0
6.10 C, I = π/6 6.11 C, I = 0 6.12 C, I = 0
6.13 D, I = ∞ 6.14 D, I = ∞ 6.18 D, ρ = 2
6.19 C, ρ = 3/4 6.20 C, ρ = 0 6.21 D, ρ = 5/4
6.22 C, ρ = 0 6.23 D, ρ = ∞ 6.24 D, ρ = 9/8
6.25 C, ρ = 0 6.26 C, ρ = (e/3)3 6.27 D, ρ =P100
6.28 C, ρ =P 4/27 6.29 D, ρ =P2 6.31 D, cf. P n−1
6.32 D, cf. n−1 6.33 C, cf. 2−n 6.34 C, cf. n−2
P −2 P −1/2
6.35 C, cf. n 6.36 D, cf. n
1
,Chapter 1 2
7.1 C 7.2 D 7.3 C 7.4 C
7.5 C 7.6 D 7.7 C 7.8 C
P −1
9.1 D, cf. n 9.2 D, an 6→ 0 P −1
9.3 C, I =P0 9.4 D, I = ∞, or cf. n
9.5 C, cf. n−2 9.6 C, ρ = 1/4
9.7 D, ρ = 4/3 9.8 C, ρ = 1/5
9.9 D, ρ = e 9.10 D, an 6→
P 0 −2
D, I = ∞, or cf.P n−1
P
9.11 9.12 C, cf. n
9.13 C, I = 0, or cf. n−2 9.14 C, alt.Pser.
9.15 D, ρ = ∞, an 6→ 0 9.16 C, cf. n−2
9.17 C, ρ = 1/27 9.18 C, alt. ser.
9.19 C 9.20 C
9.21 C, ρ = 1/2
9.22 (a) C (b) D (c) k > e
10.1 |x| < √ 1 10.2 |x| < 3/2 10.3 |x| ≤ 1
10.4 |x| ≤ 2 10.5 All x 10.6 All x
10.7 −1 ≤ x < 1 10.8 −1 < x ≤ 1 10.9 |x| < 1
10.10 |x| ≤ 1 10.11 −5 ≤ x < 5 10.12 |x| < 1/2
10.13 −1 < x ≤ 1 10.14 |x| < 3 10.15 −1 < x < 5
10.16 −1 < x < 3 10.17 −2 < x ≤ 0 10.18 −3/4 ≤ x ≤ −1/4
10.19 |x| < 3 10.20 All x 10.21 0 ≤ x √≤1
10.22 No x 10.23 x > 2 or x < −4 10.24 |x| < 5/2
10.25 nπ − π/6 < x < nπ + π/6
(−1)n (2n − 1)!!
−1/2 −1/2
13.4 = 1; =
0 n (2n)!!
Answers to part (b), Problems 5 to 19:
∞ n+2 ∞
X x X 1/2 n+1
13.5 − 13.6 x (see Example 2)
1
n 0
n
∞ ∞
(−1)n x2n
X X −1/2
13.7 13.8 (−x2 )n (see Problem 13.4)
0
(2n + 1)! 0
n
∞ ∞
X X (−1)n x4n+2
13.9 1 + 2 xn 13.10
1 0
(2n + 1)!
∞ n n ∞
X (−1) x X (−1)n x4n+1
13.11 13.12
0
(2n + 1)! 0
(2n)!(4n + 1)
∞ n 2n+1 ∞
X (−1) x X x2n+1
13.13 13.14
0
n!(2n + 1) 0
2n + 1
∞
x2n+1
X −1/2
13.15 (−1)n
0
n 2n + 1
∞ 2n ∞
X x X xn
13.16 13.17 2
0
(2n)! n
oddn
∞
X (−1)n x2n+1 ∞
X −1/2 x2n+1
13.18 13.19
0
(2n + 1)(2n + 1)! 0
n 2n + 1
2 3 5 6
13.20 x + x + x /3 − x /30 − x /90 · · ·
13.21 x2 + 2x4 /3 + 17x6 /45 · · ·
13.22 1 + 2x + 5x2 /2 + 8x3 /3 + 65x4 /24 · · ·
13.23 1 − x + x3 − x4 + x6 · · ·
,Chapter 1 3
13.24 1 + x2 /2! + 5x4 /4! + 61x6 /6! · · ·
13.25 1 − x + x2 /3 − x4 /45 · · ·
13.26 1 + x2 /4 + 7x4 /96 + 139x6 /5760 · · ·
13.27 1 + x + x2 /2 − x4 /8 − x5 /15 · · ·
13.28 x − x2 /2 + x3 /6 − x5 /12 · · ·
13.29 1 + x/2 − 3x2 /8 + 17x3 /48 · · ·
13.30 1 − x + x2 /2 − x3 /2 + 3x4 /8 − 3x5 /8 · · ·
13.31 1 − x2 /2 − x3 /2 − x4 /4 − x5 /24 · · ·
13.32 x + x2 /2 − x3 /6 − x4 /12 · · ·
13.33 1 + x3 /6 + x4 /6 + 19x5 /120 + 19x6 /120 · · ·
13.34 x − x2 + x3 − 13x4 /12 + 5x5 /4 · · ·
13.35 1 + x2 /3! + 7x4 /(3 · 5!) + 31x6 /(3 · 7!) · · ·
13.36 u2 /2 + u4 /12 + u6 /20 · · ·
13.37 −(x2 /2 + x4 /12 + x6 /45 · · · )
13.38 e(1 − x2 /2 + x4 /6 · · · )
4
13.39 1 − (x − π/2)2 /2! + (x − π/2) /4! · · ·
3
13.40 1 − (x − 1) + (x − 1)2 − (x − 1) · · ·
13.41 e [1 + (x − 3) + (x − 3) /2! + (x − 3)3 /3! · · · ]
3 2
2
13.42 −1 + (x − π) /2! − (x − π)4 /4! · · ·
13.43 −[(x − π/2) + (x − π/2)3 /3 + 2(x − π/2)5 /15 · · · ]
13.44 5 + (x − 25)/10 − (x − 25)2 /103 + (x − 25)3 /(5 · 104 ) · · ·
14.6 Error < (1/2)(0.1)2 ÷ (1 − 0.1) < 0.0056
14.7 Error < (3/8)(1/4)2 ÷ (1 − 14 ) = 1/32
14.8 For x < 0, error < (1/64)(1/2)4 < 0.001
For x > 0, error < 0.001 ÷ (1 − 12 ) = 0.002
1
14.9 Term n + 1 is an+1 = (n+1)(n+2) , so Rn = (n + 2)an+1 .
14.10 S4 = 0.3052, error < 0.0021 (cf. S = 1 − ln 2 = 0.307)
15.1 −x4 /24 − x5 /30 · · · ' −3.376 × 10−16
15.2 x8 /3 − 14x12 /45 · · · ' 1.433 × 10−16
15.3 x5 /15 − 2x7 /45 · · · ' 6.667 × 10−17
15.4 x3 /3 + 5x4 /6 · · · ' 1.430 × 10−11
15.5 0 15.6 12 15.7 10!
15.8 1/2 15.9 −1/6 15.10 −1
15.11 4 15.12 1/3 15.13 −1
15.14 t − t3 /3, error < 10−6 15.15 23 t3/2 − 52 t5/2 , error < 17 10−7
15.16 e2 − 1 15.17 √cos π2 = 0
15.18 ln 2 15.19 2
15.20 (a) 1/8 (b) 5e (c) 9/4
15.21 (a) 0.397117 (b) 0.937548 (c) 1.291286
15.22 (a) π 4 /90 (b) 1.202057 (c) 2.612375
15.23 (a) 1/2 (b) 1/6 (c) 1/3 (d) −1/2
15.24 (a) −π (b) 0 (c) −1
(d) 0 (e) 0 (f) 0
15.27 (a) 1 − vc = 1.3 × 10−5 , or v = 0.999987c
(b) 1 − vc = 5.2 × 10−7
(c) 1 − vc = 2.1 × 10−10
(d) 1 − vc = 1.3 × 10−11
15.28 mc2 + 21 mv 2
15.29 (a) F/W = θ + θ3 /3 · · ·
(b) F/W = x/l + x3 /(2l3 ) + 3x5 /(8l5 ) · · ·
, Chapter 1 4
15.30 (a) T = F (5/x + x/40 − x3 /16000 · · · )
(b) T = 21 (F/θ)(1 + θ2 /6 + 7θ4 /360 · · · )
15.31 (a) finite (b) infinite
16.1 (c) overhang: 2 3 10 100
books needed: 32 228 2.7 × 108 4 × 1086
P −3/2
16.4 C, ρ = 0 16.5 D, an 6→ P 0 −1 16.6 C, cf. n
16.7 D, I = ∞ 16.8 D, cf. n 16.9 −1 ≤ x < 1
16.10 |x| < 4 16.11 |x| ≤ 1 16.12 |x| < 5
16.13 −5 < x ≤ 1
16.14 1 − x2 /2 + x3 /2 − 5x4 /12 · · ·
16.15 −x2 /6 − x4 /180 − x6 /2835 · · ·
16.16 1 − x/2 + 3x2 /8 − 11x3 /48 + 19x4 /128 · · ·
16.17 1 + x2 /2 + x4 /4 + 7x6 /48 · · ·
16.18 x − x3 /3 + x5 /5 − x7 /7 · · ·
16.19 −(x − π) + (x − π)3 /3! − (x − π)5 /5! · · ·
16.20 2 + (x − 8)/12 − (x − 8)2 /(25 · 32 ) + 5(x − 8)3 /(28 · 34 ) · · ·
16.21 e[1 + (x − 1) + (x − 1)2 /2! + (x − 1)3 /3! · · · ]
16.22 arc tan 1 = π/4 16.23 1 − (sinπ)/π = 1
16.24 eln 3 − 1 = 2 16.25 −2
16.26 −1/3 16.27 2/3
16.28 1 16.29 6!
16.30 (b) For N = 130, 10.5821 < ζ(1.1) < 10.5868
16.31 (a) 10430 terms. For N = 200, 100.5755 < ζ(1.01) < 100.5803
16.31 (b) 2.66 × 1086 terms. For N = 15, 1.6905 < S < 1.6952
200 86
16.31 (c) ee = 103.1382×10 terms. For N = 40, 38.4048 < S < 38.4088
