cal15 - choi (kc24547) – HW15 – BERG – (56525) 1 This...

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Unformatted text preview: choi (kc24547) – HW15 – BERG – (56525) 1 This print-out should have 14 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Compare the radius of convergence, R 1 , of the series ∞ summationdisplay n =0 c n x n with the radius of convergence, R 2 , of the series ∞ summationdisplay n = 1 n c n x n − 1 when lim n →∞ vextendsingle vextendsingle vextendsingle c n +1 c n vextendsingle vextendsingle vextendsingle = 3 . 1. R 1 = 2 R 2 = 3 2. R 1 = 2 R 2 = 1 3 3. 2 R 1 = R 2 = 3 4. R 1 = R 2 = 3 5. 2 R 1 = R 2 = 1 3 6. R 1 = R 2 = 1 3 correct Explanation: When lim n →∞ vextendsingle vextendsingle vextendsingle c n +1 c n vextendsingle vextendsingle vextendsingle = 3 , the Ratio Test ensures that the series ∞ summationdisplay 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 →∞ vextendsingle vextendsingle vextendsingle ( n + 1) c n +1 nc n vextendsingle vextendsingle vextendsingle = lim n →∞ vextendsingle vextendsingle vextendsingle c n +1 c n vextendsingle vextendsingle vextendsingle , the Ratio Test ensures also that the series ∞ summationdisplay 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 . 002 10.0 points Find a power series representation for the function f ( t ) = 1 2 + t . 1. f ( t ) = ∞ summationdisplay n =0 1 2 n +1 t n 2. f ( t ) = ∞ summationdisplay n =0 (- 1) n 2 n +1 t n 3. f ( t ) = ∞ summationdisplay n =0 (- 1) n 2 n +1 t n correct 4. f ( t ) = ∞ summationdisplay n =0 (- 1) n 2 t n 5. f ( t ) = ∞ summationdisplay n =0 2 n +1 t n Explanation: We know that 1 1- x = 1 + x + x 2 + . . . = ∞ summationdisplay n =0 x n . On the other hand, 1 2 + t = 1 2 parenleftBig 1 1- (- t/ 2) parenrightBig . choi (kc24547) – HW15 – BERG – (56525) 2 Thus f ( t ) = 1 2 ∞ summationdisplay n =0 parenleftbigg- t 2 parenrightbigg n = 1 2 ∞ summationdisplay n = 0 (- 1) n 2 n t n . Consequently, f ( t ) = ∞ summationdisplay n = 0 (- 1) n 2 n +1 t n with | t | < 2. 003 10.0 points Find a power series representation centered at the origin for the function f ( y ) = y 3 (6- y ) 2 . 1. f ( y ) = ∞ summationdisplay n = 3 n 6 n y n 2. f ( y ) = ∞ summationdisplay n = 2 1 6 n − 1 y n 3. f ( y ) = ∞ summationdisplay n = 3 n- 2 6 n − 1 y n correct 4. f ( y ) = ∞ summationdisplay n = 3 1 6 n − 3 y n 5. f ( y ) = ∞ summationdisplay n = 2 n- 1 6 n y n Explanation: By the known result for geometric series, 1 6- y = 1 6 parenleftBig 1- y 6 parenrightBig = 1 6 ∞ summationdisplay n =0 parenleftBig y 6 parenrightBig n = ∞ summationdisplay n = 0 1 6 n +1 y n ....
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This note was uploaded on 04/04/2011 for the course MATH 408 L taught by Professor Zheng during the Spring '10 term at University of Texas at Austin.

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cal15 - choi (kc24547) – HW15 – BERG – (56525) 1 This...

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