p8 - investigate the behavior (convergence or divergence)...

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Math 5615H: Introduction to Analysis I. Fall 2011 Homework #8 (due on Wednesday, November 2). 50 points are divided between 4 problems. #1. (12 points). Let 0 < x 1 = a < x 2 = b be arbitrary real number, and let x n := 1 2 ( x n - 2 + x n - 1 ) for n = 3 , 4 , 5 ,.... Show that the sequence { x n } converges, and find its limit. #2. (12 points, Exercise 4 on p. 78). Find the upper and lower limits of the sequence { s n } defined by s 0 = 0; s 2 m = s 2 m - 1 2 ; s 2 m +1 = 1 2 + s 2 m . #3. (12 points). Using the fact that lim n →∞ 1 + 1 n · n = e = 2 . 71828 ...,
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Unformatted text preview: investigate the behavior (convergence or divergence) of the sequences a n := n n n ! 3 n and b n := n n n ! 2 n , where n ! := 1 2 3 ( n-1) n. #4. (14 points, Exercise 17(a,c) on p.81). Fix &gt; 1. Take x 1 &gt; and dene x n +1 := + x n 1 + x n for n = 1 , 2 , 3 ,.... (a). Prove that x 1 &gt; x 3 &gt; x 5 &gt; . . (c). Prove that lim x n = ....
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