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# s9 - Math 5615H Introduction to Analysis I Homework#9...

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Math 5615H: Introduction to Analysis I. Fall 2011 Homework #9. Problems and Solutions. #1. 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. Solution. We have 1 n ( n + 1)( n + 2) = 1 / 2 n + - 1 n + 1 + 1 / 2 n + 2 . Then the n th partial sum s n = n k =1 ( 1 / 2 k + - 1 k + 1 + 1 / 2 k + 2 ) = ( 1 / 2 1 + - 1 2 + 1 / 2 3 ) + ( 1 / 2 2 + - 1 3 + 1 / 2 4 ) + · · · + ( 1 / 2 n - 1 + - 1 n + 1 / 2 n + 1 ) + ( 1 / 2 n + - 1 n + 1 + 1 / 2 n + 2 ) = 1 4 + - 1 / 2 n + 1 + 1 / 2 n + 2 s = 1 4 as n → ∞ . In other words, the sum of this series s = 1 / 4. #2. 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 . Solution. (i). By the elementary inequality ab ( a 2 + b 2 ) / 2 with a = a n and b = n - c , we have b n := a n n c 1 2 ( a n + n - 2 c ) .

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