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hw4 - Exercise 3 Consider the power series expansion 1 1-2...

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Putnam Exam Seminar Assignment 4 Fall 2010 Due October 4 Exercise 1. The sequence { a n } is defined by a 1 = 1, a 2 = 2, a 3 = 24, and, for n 4, a n = 6 a 2 n - 1 a n - 3 - 8 a n - 1 a 2 n - 2 a n - 2 a n - 3 . Show that, for all n , a n is an integer multiple of n . [Putnam Exam, 1999, A-6] Exercise 2. Let { x n } be a sequence of nonzero real numbers such that x 2 n - x n - 1 x n +1 = 1 for n 1. Prove that there is a real number a so that x n +1 = ax n - x n - 1 for all n 1. [Putnam Exam, 1993, A-2]
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Unformatted text preview: Exercise 3. Consider the power series expansion 1 1-2 x-x 2 = ∞ X n =0 a n x n . Prove that, for each integer n ≥ 0, there is an integer m such that a 2 n + a 2 n +1 = a m . [Putnam Exam, 1999, A-3] Exercise 4. Evaluate lim x → 1-∞ Y n =0 ± 1 + x n +1 1 + x n ² x n . [Putnam Exam, 2004, B-5]...
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