Math 301 Problem Set 6 Solutions

Math 301 Problem Set 6 Solutions - Math 301/ENAS 513...

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Unformatted text preview: Math 301/ENAS 513: Homework 6 Solution Triet Le 1. Page 66: 2. Determine the coefficients { a n } ∞ n =0 of the power series whose sum is (1- z )- 2 . Solution 1 . Let f ( z ) = (1- z )- 2 = ∑ ∞ n =0 a n z n . Then a n = f ( n ) (0) n ! , where f ( n ) (0) = ( n + 1)! . 2. Page 66: 4. Solution 2 . Use Theorem 5.4, and induct on k . 3. Page 66: 5.Suppose that ∑ ∞ n =0 a n z n has radius of convergence R > 0, and suppose that | z | = r < R . Show that there exists a power series ∑ ∞ n =0 b n z n with the radius of convergence at least R- r such that ∞ X n =0 a n z n = ∞ X n =0 b n ( z- z ) n , | z- z | < R- r. Solution 3 . Suppose ∑ ∞ n =0 a n z n has radius of convergence R > 0, and suppose | z | = r < R . Define g ( z ) = ∞ X n =0 a n ( z + z ) n , | z | < R- r. Clearly, ∑ ∞ n =0 a n ( z + z ) n converges absolutely for all | z | < R- r . Now expanding ( z + z ) n for all n and group the coefficients in front of z n together to obtain a new series ∑ ∞ n =0...
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Math 301 Problem Set 6 Solutions - Math 301/ENAS 513...

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