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# hw8 - X n =1 a n n a>(ii Explain how to use the result of...

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CALCULUS 1501 WINTER 2010 HOMEWORK ASSIGNMENT 8. Due March 12. 8.1. Determine whether the series converges absolutely, conditionally, or diverges. (i) X n =1 ( - 1) n - 1 n n + 10 (ii) X n =1 n 5 2 - 3 n 4 n + 3 n (iii) X n =1 sin 5 n n 5 . 8.2. Let { f n } be the Fibonacci sequence, given by f 1 = f 2 = 1, f n = f n - 1 + f n - 2 , for n > 2. Use Problem 5.3 ( Homework 5 ) to assess the convergence of the series X n =1 1 f n 8.3. Determine the convergence of the series 2 5 + 2 · 6 5 · 8 + 2 · 6 · 10 5 · 8 · 11 + 2 · 6 · 10 · 14 5 · 8 · 11 · 14 + · · · 8.4. (i) Show that the following series converges.
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Unformatted text preview: X n =1 a n n ! , a > . (ii) Explain how to use the result of part (i) to prove that lim n →∞ a n n ! = 0 for all a > 0. 8.5. Determine whether the series below converges absolutely, conditionally, or diverges. ∞ X n =1 (-1) n-1 3 n n ! n n Hint: Recall that lim n →∞ ± 1 + 1 n ² n = e . 8.6. Assess the convergence of the following alternating series ∞ X n =2 ± 1 √ n-1-1 √ n + 1 ² . 1...
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