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Unformatted text preview: Homework 8 Math CS 120, Winter 2009 Due on Thursday, March 12th, 2009 1. Investigate the convergence of a n where (a) a n = n + 1 n n + 1 . (b) a n = s n + 1 n n + 1 . 2. Let a n and b n converge, with b n &gt; 0 for all n . Suppose that a n /b n L . Prove that lim N n = N a n n = N b n = L. 3. Let { a n } 0. Show that a n converges if and only if 2 n a 2 n con verges. 4. Show that if f 0 and if f is monotonically decreasing, and if c n = n X k =1 f ( k ) Z n 1 f ( x ) dx, then lim n c n exists. 5. Show that if c n 0 and c n converges, c n /n also converges. 6. Estimate n X k =1 k. 7. Let n ( x ) be positivevalued and continuous for all x [ 1 , 1], with lim n Z 1 1 n ( x ) dx = 1 . 1 2 Suppose that { n } converges to 0 uniformly on the intervals [ 1 , c ] and [ c, 1] for any c &gt; 0. Let g be a continuous function on [ 1 , 1]....
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This document was uploaded on 12/27/2011.
 Fall '09
 Math

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