fr - Final Review Problems 1 Given an(x c)n f(x = n=0 1...

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Final Review Problems November 27, 2007 1. Given f ( x ) = X n =0 a n ( x - c ) n , let R = 1 lim n →∞ n p | a n | . Explain why f converges absolutely for all x such that | x - c | < R and diverges for all x such that | x - c | > R . 2. Let p, q be real numbers with p < q . Find a power series whose interval of convergence is (a) ( p, q ) (b) ( p, q ] (c) [ p, q ) (d) [ p, q ] 3. Suppose c n x n converges when x = - 4 and diverges when x = 6. What can be said about the convergence or divergence of (a) c n (b) c n 8 n (c) c n ( - 3) n (d) ( - 1) n c n 9 n 4. Suppose c n x n has radius of convergence 2 and d n x n has radius of convergence 3. What is the radius of convergence of ( c n + d n ) x n ? 5. Find a power series representation of (a) f ( x ) = 1 (1 + x ) 2 (b) g ( x ) = 1 (1 + x ) 3 (c) h ( x ) = x 2 (1 + x ) 3 6. Approximate Z 1 0 x cos x 3 dx to within 10 - 3 . 7.

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This note was uploaded on 10/18/2011 for the course MATH V1102 taught by Professor Mosina during the Fall '08 term at Columbia.

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fr - Final Review Problems 1 Given an(x c)n f(x = n=0 1...

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