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hw6-extra - a find the value of this series 3(a Prove that...

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Homework 6: Additional Problems (required). Advanced Calculus I. Math 451, Fall 2011 1. Prove that each of the following series converge and find its value. (a) X k =1 ( - 1) k +1 π k (b) X k =1 ( - 1) k + 4 5 k . (Hint: split up into two series, for odd and for even terms). 2. Find all real numbers a for which the series X n =1 ( a n - a n - 1 )( a n + a n - 1 ) converges. For each such
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Unformatted text preview: a , find the value of this series. 3. (a) Prove that if ∑ n a n converges then the sequence of its partial sums s n is bounded. (b) Show that the converse of part (a) is false. Namely, find a series ∑ n a n with bounded partial sums but which diverges....
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