104hw5sum06 - MATH 104, SUMMER 2006, HOMEWORK 5 SOLUTION...

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MATH 104, SUMMER 2006, HOMEWORK 5 SOLUTION BENJAMIN JOHNSON Due July 19 Assignment: Section 14: 14.4, 14.7, 14.10, 14.13(d) Section 15: 15.4, 15.6, 15.7 Section 14 14.4 Determine which of the following series converge. Justify your answer. (a) n = 2 1 [ n + ( - 1) n ] 2 The series converges. I’ll use the comparison test. We have ± ± ± ± 1 [ n + ( - 1) n ] 2 ± ± ± ± 1 ( n - 1) 2 for every n 2. Since n = 2 1 ( n - 1) 2 = n = 1 1 n 2 converges, so does n = 2 1 [ n + ( - 1) n ] 2 by the comparison test. (b) [ n + 1 - n ] The series diverges. By the telescoping sum formula, N j = 0 [ n + 1 - n ] = N + 1 - 0. So j = 0 [ n + 1 - n ] = lim n →∞ N j = 0 [ n + 1 - n ] = lim n →∞ N + 1 = . (c) n ! n n The series converges. I’ll use the ratio test. We have ± ± ± ± ± ± ± ( n + 1)! ( n + 1) n + 1 n ! n n ± ± ± ± ± ± ± = ² n n + 1 ³ n = ² n n + 1 ³ n + 1 · n + 1 n = ´ 1 - 1 n + 1 ! n + 1 · n + 1 n Taking the limit as n → ∞ and applying the limit product law yields 1 e · 1 = 1 e < 1. So the series converges by the ratio test. 14.7 Prove that if
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104hw5sum06 - MATH 104, SUMMER 2006, HOMEWORK 5 SOLUTION...

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