AnsHW13 - Cheung, Anthony Homework 13 Due: Nov 28 2006,...

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Unformatted text preview: Cheung, Anthony Homework 13 Due: Nov 28 2006, 3:00 am Inst: David Benzvi 1 This print-out should have 11 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. The due time is Central time. 001 (part 1 of 1) 10 points Find the interval of convergence of the se- ries X n = 1 x n n + 5 . 1. interval of cgce = (- 5 , 5] 2. interval of cgce = [- 5 , 5] 3. interval of cgce = [- 1 , 1] 4. interval of cgce = [- 1 , 1) correct 5. interval of cgce = (- 1 , 1) 6. converges only at x = 0 Explanation: When a n = x n n + 5 , then fl fl fl fl a n +1 a n fl fl fl fl = fl fl fl fl x n +1 n + 6 n + 5 x n fl fl fl fl = | x | n + 5 n + 6 = | x | r n + 5 n + 6 . But lim n n + 5 n + 6 = 1 , so lim n fl fl fl fl a n +1 a n fl fl fl fl = | x | . By the Ratio Test, therefore, the given series (i) converges when | x | < 1, (ii) diverges when | x | > 1. We have still to check what happens at the endpoints x = 1. At x = 1 the series becomes ( * ) X n = 1 1 n + 5 . Applying the Integral Test with f ( x ) = 1 x + 5 we see that f is continuous, positive, and de- creasing on [1 , ), but the improper integral I = Z 1 f ( x ) dx diverges, so the infinite series ( * ) diverges also. On the other hand, at x =- 1, the series becomes ( ) X n = 1 (- 1) n n + 5 . which is an alternating series X n = 1 (- 1) n a n , a n = f ( n ) with f ( x ) = 1 x + 5 the same continuous, positive and decreasing function as before. Since lim x f ( x ) = lim x 1 x + 5 = 0 , however, the Alternating Series Test ensures that ( ) converges. Consequently, the interval of convergence = [- 1 , 1) . keywords: power series, interval of conver- gence, ratio test, integral test, alternating series test Cheung, Anthony Homework 13 Due: Nov 28 2006, 3:00 am Inst: David Benzvi 2 002 (part 1 of 1) 10 points Find the interval of convergence of the se- ries X n = 1 (- 1) n x n 3 n 2 + 4 . 1. interval of cgce = [- 1 , 1) 2. interval of cgce = [- 1 , 1] correct 3. interval of cgce = (- 3 , 4] 4. interval of cgce = [- 3 , 4] 5. interval of cgce = (- 1 , 1] 6. converges only at x = 0 Explanation: When a n = (- 1) n x n 3 n 2 + 4 , then fl fl fl fl a n +1 a n fl fl fl fl = fl fl fl fl- x n +1 3( n + 1) 2 + 4 3 n 2 + 4 x n fl fl fl fl = | x | 3 n 2 + 4 3( n + 1) 2 + 4 . But 3( n + 1) 2 + 4 = 3 n 2 + 6 n + 7 , while lim n 3 n 2 + 4 3 n 2 + 6 n + 7 = 1 ....
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AnsHW13 - Cheung, Anthony Homework 13 Due: Nov 28 2006,...

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