p9 - a n , s n = a 1 + + a n , and a n diverges. (a). Prove...

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Math 5615H: Introduction to Analysis I. Fall 2011 Homework #9 (due on Wednesday, November 9). 50 points are divided between 4 problems. #1. (10 points). Using partial-fraction decomposition 1 n ( n + 1)( n + 2) = A n + B n + 1 + C n + 2 , show that the series n =1 1 n ( n + 1)( n + 2) converges and find its sum. #2. (14 points). Find a constant c 0 > 0 satisfying the following properties. (i). The convergence of every series ± a n with a n 0 implies the convergence of n =1 b n := n =1 a n n c for each c > c 0 . (ii). There is a convergent series ± a n with a n 0, such that the series ± b n diverges if c = c 0 . #3. (14 points, Exercise 11(a,d) on p.79). Suppose
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Unformatted text preview: a n , s n = a 1 + + a n , and a n diverges. (a). Prove that n =1 b n := n =1 a n 1 + a n diverges . (d). What can be said about n =1 c n := n =1 a n 1 + na n and n =1 d n := n =1 a n 1 + n 2 a n ? #4. (12 points). Using two-sided estimate 1 n + 1 < ln ( 1 + 1 n ) < 1 n , show that the sequence s n = 1 + 1 2 + 1 3 + + 1 n-ln n, where ln x := log e x, is convergent....
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