Mathematical Methods in the Physical Sciences (3rd Edition) by Mary L. Boas – Solution Manual | Complete Worked Solutions for All Chapters

,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 1 (−1)n+1 (−1)n
4.7 an = (−1) n+1 + → 0 ; Sn = 1 + → 1; Rn = n + 1 → 0
n 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
Note: In the following answers, I = R ∞ 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 = 100
6.28 C, ρ =P 6.29 D, ρ = 2 6.31 D, cf.
4/27 P n−2
−1
P
6.32 D, cf. P n−1 6.33 C, cf. P 2−n 6.34 C, cf. n
6.35 C, cf. n−2 6.36 D, cf. n−1/2




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
9.1 D, cf. n−1 9.2 D, an 6→ 0 P −1
9.3 C, I = P
0 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 →
P P 0 −2
9.11 D, I = ∞, or cf.P n−1 9.12 C, cf. n
9.13 C, I = 0, or cf. n−2 9.14 C, alt. P
ser.
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

−1/2 −1/2 (−1)n(2n − 1)!!
13.4 = 1; =
0 n (2n)!!
Answers to part (b), Problems 5 to 19:
∞ ∞
X xn+2 X 1/2 n+1
13.5 − 13.6 x (see Example 2)
1
n 0
n
∞ ∞
X (−1) nx2n 13.8
X −1/2
13.7 (−x2)n (see Problem 13.4)
(2n + 1)! n
0 ∞ 0
∞ n 4n+2
13.9 X X (−1) x
1+2 xn 13.10
(2n + 1)!
∞ 1 n n 0
∞ n 4n+1
X (−1) x X (−1) x
13.11 (2n + 1)! 13.12 (2n)!(4n + 1)
∞0 n 2n+1 0
∞ 2n+1
X (−1) x Xx
13.13 n!(2n + 1) 13.14
∞
0 2n + 1
X −1/2 0
n x2n+1
13.15 (−1)
n 2n + 1
0
∞ 2n ∞
n
X x Xx
13.16 13.17 2
0
(2n)! oddn
n
X
∞ ∞
13.18 (−1)nx2n+1 X −1/2 x2n+1
13.19
(2n + 1)(2n + 1)! n 2n + 1
0 0
13.20 x + x2 + x3/3 − x5/30 − x6/90 · · ·
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 e3[1 + (x − 3) + (x − 3)2/2! + (x − 3)3/3! · · · ]
13.42 −1 + (x − π)2/2! − (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
14.9 Term n + 1 is an+1 = (n+1)(n+2)
1 , 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 2 t3/2 − 2 t5/2, error < 1 10−7
3 5 7
15.16 e2 − 1 15.17 cos π = 0
√ 2
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) π4/90 (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 + 12 mv2
15.29 (a) F/W = θ + θ3/3 · · ·
(b) F/W = x/l + x3/(2l3) + 3x5/(8l5) · · ·