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Unformatted text preview: Math 3110 Spring 09 Homework 6: Selected Solutions [8.2-1 (a, c, e)]: Claim : (a) X 1 x n 2 n n diverges at the right endpoint of its radius of con- vergence R = 2 and converges at the left endpoint of this interval. (b) X 1 x n n n diverges at both endpoints of its radius of convergence R = 1. (e) X 1 n + 2 n n x n diverges at both endpoints of its radius of convergence R = 1 Proof (a) Use the ratio test to determine that R = 2: lim n fl fl fl fl fl fl fl fl x n +1 2 n +1 n + 1 x n 2 n n fl fl fl fl fl fl fl fl = 1 2 lim n r n n + 1 | x | = 1 2 lim n r 1- 1 n + 1 ! | x | < 1 when | x | < 2. At x = 2, X 1 x n 2 n n = X 1 1 n , which diverges by comparison with the divergent p-series X 1 1 n . When x =- 2, we have an alternating series X 1 (- 1) n 1 n that converges by Cauchys Test since its terms a n = (- 1) n- 1 1 n go to zero in absolute value....
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This note was uploaded on 11/09/2009 for the course MATH 3110 taught by Professor Ramakrishna during the Fall '08 term at Cornell University (Engineering School).
- Fall '08