Mathematic Methods HW Solutions 2

# Mathematic Methods HW Solutions 2 - Chapter 1 2 7.1 7.5 C C...

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Unformatted text preview: 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 < 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 + 1 x2n (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 ...
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