t3_s2_sln - YORK UNIVERSITY Faculty of Pure and Applied...

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Faculty of Pure and Applied Science AS/SC/MATH 1014 3.0 M June - August 2003 Term Test 3 SOLUTIONS 1. (4 points) Which of the following statements about series is true? (a) If lim k →∞ u k = 0 , then Σ n =1 u k converges (b) If lim k →∞ u k 6 = 0 , then Σ n =1 u k diverges (c) If Σ n =1 u k diverges, then lim k →∞ u k 6 = 0 (d) Σ n =1 u k converges, if and only if lim k →∞ u k = 0 (e) None of the preceding. Answer: (b). 2. (9 points) Determine whether the given series converges or diverges. Justify your answers by indicating a relevant test. (a) Σ n =1 e n n ! Answer: ρ = lim n →∞ a n +1 a n = lim n →∞ e n +1 ( n +1)! e n n ! = lim n →∞ e n + 1 = 0 < 1 . Hence, the series is convergent by the Ratio Test. (b) Σ n =1 1 2 n - 1 Answer: Denote Σ n =1 1 2 n - 1 = Σ n =1 a n . Compare the series with the p -series Σ n =1 b n = Σ n =1 1 n , which is divergent since p = 1 2 < 1 . lim n →∞ a n b n = lim n →∞ 1 2 n - 1 1 n = lim n →∞ n 2 n - 1 = 1 2 < . Hence, by the Limit Comparison Test, the series is divergent too.
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This note was uploaded on 05/18/2010 for the course MATH 1014 taught by Professor Ganong during the Spring '09 term at York University.

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t3_s2_sln - YORK UNIVERSITY Faculty of Pure and Applied...

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