HW 14 key

# HW 14 key - Miller Kierste Homework 14 Due May 1 2007 3:00...

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Miller, Kierste – Homework 14 – Due: May 1 2007, 3:00 am – Inst: Gary Berg 1 This print-out should have 18 questions. Multiple-choice questions may continue on the next column or page – fnd 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 x n with the radius oF convergence, R 2 , oF the series X n = 1 n c n x n - 1 when lim n →∞ f f f c n +1 c n f f f = 3 . 1. R 1 = 2 R 2 = 1 3 2. R 1 = R 2 = 3 3. R 1 = R 2 = 1 3 correct 4. 2 R 1 = R 2 = 1 3 5. R 1 = 2 R 2 = 3 6. 2 R 1 = R 2 = 3 Explanation: When lim n →∞ f f f c n +1 c n f f f = 3 , the Ratio Test ensures that the series X n = 0 c n x n is (i) convergent when | x | < 1 3 , and (ii) divergent when | x | > 1 3 . On the other hand, since lim n →∞ f f f ( n + 1) c n +1 nc n f f f = lim n →∞ f f f c n +1 c n f f f , the Ratio Test ensures also that the series X n = 1 n c n x n - 1 is (i) convergent when | x | < 1 3 , and (ii) divergent when | x | > 1 3 . Consequently, R 1 = R 2 = 1 3 . keywords: 002 (part 1 oF 1) 10 points ±ind a power series representation For the Function f ( y ) = 1 2 + y . 1. f ( y ) = X n = 0 ( - 1) n 2 y n 2. f ( y ) = X n = 0 1 2 n +1 y n 3. f ( y ) = X n = 0 ( - 1) n 2 n +1 y n correct 4. f ( y ) = X n = 0 2 n +1 y n 5. f ( y ) = X n = 0 ( - 1) n 2 n +1 y n Explanation: We know that 1 1 - x = 1 + x + x 2 + . . . = X n = 0 x n .

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Miller, Kierste – Homework 14 – Due: May 1 2007, 3:00 am – Inst: Gary Berg 2 On the other hand, 1 2 + y = 1 2 1 1 - ( - y/ 2) · . Thus f ( y ) = 1 2 X n = 0 - y 2 · n = 1 2 X n = 0 ( - 1) n 2 n y n . Consequently, f ( y ) = X n = 0 ( - 1) n 2 n +1 y n with | y | < 2. keywords: 003 (part 1 of 1) 10 points Find a power series representation for the function f ( x ) = x 4 x + 1 . 1. f ( x ) = X n = 0 ( - 1) n 2 n x n +1 2. f ( x ) = X n = 0 2 2 n x n +1 3. f ( x ) = X n = 0 2 n x n 4. f ( x ) = X n = 0 ( - 1) n 2 2 n x n +1 correct 5. f ( x ) = X n = 0 2 2 n x n 6. f ( x ) = X n = 0 ( - 1) n 2 n x n Explanation: After simpli±cation, f ( x ) = x 4 x + 1 = x 1 - ( - 4 x ) . On the other hand, 1 1 - x = X n = 0 x n . Thus f ( x ) = x n X n = 0 ( - 4 x ) n o = x n X n = 0 ( - 1) n 2 2 n x n . Consequently, f ( x ) = X n = 0 ( - 1) n 2 2 n x n +1 . keywords: 004 (part 1 of 1) 10 points Find a power series representation centered at the origin for the function f ( x ) = x 3 (3 - x ) 2 . 1. f ( x ) = X n = 2 1 3 n - 1 x n 2. f ( x ) = X n = 3 n 3 n x n 3. f ( x ) = X n = 3 1 3 n - 3 x n 4. f ( x ) = X n = 2 n - 1 3 n x n 5. f ( x ) = X n = 3 n - 2 3 n - 1 x n correct Explanation:
Miller, Kierste – Homework 14 – Due: May 1 2007, 3:00 am – Inst: Gary Berg 3 By the known result for geometric series, 1 3 - x = 1 3 1 - x 3 · = 1 3 X n = 0 x 3 · n = X n = 0 1 3 n +1 x n . This series converges on (

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HW 14 key - Miller Kierste Homework 14 Due May 1 2007 3:00...

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