S9 - 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

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Unformatted text preview: 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 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 . (ii). There is a convergent series ∑ a n with a n ≥ 0, such that the series ∑ b n diverges if c = c ....
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S9 - 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

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