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Unformatted text preview: Kim, Jin – Homework 14 – Due: Dec 4 2007, 3:00 am – Inst: Diane Radin 1 This printout should have 17 questions. Multiplechoice 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 Compare the radius of convergence, R 1 , of the series ∞ X n = 0 c n y n with the radius of convergence, R 2 , of the series ∞ X n = 1 n c n y n 1 when lim n →∞ fl fl fl c n +1 c n fl fl fl = 2 . 1. R 1 = 2 R 2 = 2 2. R 1 = R 2 = 2 3. R 1 = R 2 = 1 2 correct 4. 2 R 1 = R 2 = 2 5. R 1 = 2 R 2 = 1 2 6. 2 R 1 = R 2 = 1 2 Explanation: When lim n →∞ fl fl fl c n +1 c n fl fl fl = 2 , the Ratio Test ensures that the series ∞ X n = 0 c n y n is (i) convergent when  y  < 1 2 , and (ii) divergent when  y  > 1 2 . On the other hand, since lim n →∞ fl fl fl ( n + 1) c n +1 nc n fl fl fl = lim n →∞ fl fl fl c n +1 c n fl fl fl , the Ratio Test ensures also that the series ∞ X n = 1 n c n y n 1 is (i) convergent when  y  < 1 2 , and (ii) divergent when  y  > 1 2 . Consequently, R 1 = R 2 = 1 2 . keywords: 002 (part 1 of 1) 10 points Find a power series representation for the function f ( x ) = 1 x 2 . 1. f ( x ) = ∞ X n = 0 2 n x n 2. f ( x ) = ∞ X n = 0 ( 1) n 1 2 n +1 x n 3. f ( x ) = ∞ X n = 0 ( 1) n 2 n x n 4. f ( x ) = ∞ X n = 0 1 2 n +1 x n 5. f ( x ) = ∞ X n = 0 1 2 n +1 x n correct Explanation: We know that 1 1 x = 1 + x + x 2 + . . . = ∞ X n = 0 x n . Kim, Jin – Homework 14 – Due: Dec 4 2007, 3:00 am – Inst: Diane Radin 2 On the other hand, 1 x 2 = 1 2 ‡ 1 1 ( x/ 2) · . Thus f ( x ) = 1 2 ∞ X n = 0 ‡ x 2 · n = 1 2 ∞ X n = 0 1 2 n x n . Consequently, f ( x ) = ∞ X n = 0 1 2 n +1 x n with  x  < 2. keywords: 003 (part 1 of 1) 10 points Find a power series representation for the function f ( x ) = ln(3 x ) . 1. f ( x ) = ln3 + ∞ X n = 0 x n n 3 n 2. f ( x ) = ln3 ∞ X n = 0 x n 3 n 3. f ( x ) = ∞ X n = 1 x n n 3 n 4. f ( x ) = ln3 ∞ X n = 1 x n n 3 n correct 5. f ( x ) = ∞ X n = 0 x n n 3 n 6. f ( x ) = ln3 + ∞ X n = 1 x n 3 n Explanation: We can either use the known power series representation ln(1 x ) = ∞ X n = 1 x n n , or the fact that ln(1 x ) = Z x 1 1 s ds = Z x n ∞ X n = 0 s n o ds = ∞ X n = 0 Z x s n ds = ∞ X n = 1 x n n . For then by properties of logs, f ( x ) = ln3 ‡ 1 1 3 x · = ln3 + ln ‡ 1 1 3 x · , so that f ( x ) = ln3 ∞ X n = 1 x n n 3 n . keywords: 004 (part 1 of 1) 10 points Find a power series representation centered at the origin for the function f ( y ) = y 3 (2 y ) 2 . 1. f ( y ) = ∞ X n = 3 n 2 2 n 1 y n correct 2. f ( y ) = ∞ X n = 2 1 2 n 1 y n 3. f ( y ) = ∞ X n = 3 n 2 n y n 4. f ( y ) = ∞ X n = 3 1 2 n 3 y n 5. f ( y ) = ∞ X n = 2 n 1 2 n y n Explanation: Kim, Jin – Homework 14 – Due: Dec 4 2007, 3:00 am – Inst: Diane Radin 3 By the known result for geometric series, 1 2 y = 1 2 ‡ 1 y 2 · = 1 2 ∞ X n = 0...
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This note was uploaded on 04/12/2010 for the course PHY 58195 taught by Professor Turner during the Spring '09 term at University of Texas.
 Spring '09
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