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hw14solutions - Math 521 Advanced Calculus I Instructor J...

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Unformatted text preview: Math 521 - Advanced Calculus I Instructor: J. Metcalfe Due: March 1, 2010 Assignment 14 1. Show that (a) ∞ X n =0 1 ( n + 1)( n + 2) = 1, (b) ∞ X n =0 1 ( α + n )( α + n + 1) = 1 α > , if α > 0, (c) ∞ X n =1 1 n ( n + 1)( n + 2) = 1 4 . To each of these, we apply partial fractions. For (a), we have 1 ( n + 1)( n + 2) = 1 n + 1- 1 n + 2 . Thus, s k = ((1- (1 / 2)) + ((1 / 2)- (1 / 3)) + ((1 / 3)- (1 / 4)) + ... + ((1 /k )- (1 /k + 1)) + ((1 /k + 1)- (1 /k + 2)) = 1- 1 k + 2 . Since s k → 1, part (a) follows. Part (b) follows similarly by noting (via partial fractions) that 1 ( n + α )( α + n + 1) = 1 n + α- 1 n + α + 1 and as such s k = 1 α- 1 k + α + 1 → 1 α . For part (c), we have 1 n ( n + 1)( n + 2) = 1 / 2 n- 1 n + 1 + 1 / 2 n + 2 and s k = (1 / 4)- 1 / 2 k + 1 + 1 / 2 k + 2 → 1 / 4 . 1 2. Let ∑ ∞ n =1 a ( n ) be such that { a ( n ) } is a decreasing sequence of strictly positive numbers. If s ( n ) denotes the n th partial sum, show (by grouping the terms in...
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hw14solutions - Math 521 Advanced Calculus I Instructor J...

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