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Hwk6_Solns

Hwk6_Solns - Math 3110 Spring ’09 Homework 6 Selected...

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Unformatted text preview: Math 3110 Spring ’09 Homework 6: Selected Solutions [8.2-1 (a, c, e)]: Claim : (a) ∞ X 1 x n 2 n √ n diverges at the right endpoint of its radius of con- vergence R = 2 and converges at the left endpoint of this interval. (b) ∞ X 1 x n n √ n diverges at both endpoints of its radius of convergence R = 1. (e) ∞ X 1 n + 2 n ¶ n x n diverges at both endpoints of its radius of convergence R = 1 Proof (a) Use the ratio test to determine that R = 2: lim n →∞ fl fl fl fl fl fl fl fl x n +1 2 n +1 √ n + 1 x n 2 n √ n fl fl fl fl fl fl fl fl = 1 2 lim n →∞ r n n + 1 ¶ | x | = 1 2 lim n →∞ ˆ r 1- 1 n + 1 ! | x | < 1 when | x | < 2. At x = 2, ∞ X 1 x n 2 n √ n = ∞ X 1 1 √ n , which diverges by comparison with the divergent p-series ∞ X 1 1 n . When x =- 2, we have an alternating series ∞ X 1 (- 1) n 1 √ n that converges by Cauchy’s Test since its terms a n = (- 1) n- 1 1 √ n go to zero in absolute value....
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Hwk6_Solns - Math 3110 Spring ’09 Homework 6 Selected...

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