HW14-solutions - cadena(jc59484 HW14 lawn(55930 This print-out should have 17 questions Multiple-choice questions may continue on the next column or

# HW14-solutions - cadena(jc59484 HW14 lawn(55930 This...

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Unformatted text preview: cadena (jc59484) – HW14 – lawn – (55930) 1 This print-out should have 17 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 z n with the radius of convergence, R 2 , of the series ∞ summationdisplay n =1 n c n z n- 1 when lim n →∞ | c n | 1 /n = 1 4 . 1. 2 R 1 = R 2 = 1 4 2. 2 R 1 = R 2 = 4 3. R 1 = R 2 = 1 4 4. R 1 = 2 R 2 = 4 5. R 1 = 2 R 2 = 1 4 6. R 1 = R 2 = 4 correct Explanation: When lim n →∞ | c n | 1 /n = 1 4 , the Root Test ensures that the series ∞ summationdisplay n =0 c n z n is (i) convergent when | z | < 4, and (ii) divergent when | z | > 4. On the other hand, since lim n →∞ | n c n | 1 /n = lim n →∞ | c n | 1 /n , the Root Test ensures also that the series ∞ summationdisplay n =1 n c n z n- 1 is (i) convergent when | z | < 4, and (ii) divergent when | z | > 4. Consequently, R = R 2 = 4 . 002 10.0 points Compare the radius of convergence, R 1 , of the series ∞ summationdisplay n = 0 c n y n with the radius of convergence, R 2 , of the series ∞ summationdisplay n =1 n c n y n- 1 when lim n →∞ vextendsingle vextendsingle vextendsingle c n +1 c n vextendsingle vextendsingle vextendsingle = 6 . 1. 2 R 1 = R 2 = 1 6 2. R 1 = 2 R 2 = 1 6 3. 2 R 1 = R 2 = 6 4. R 1 = R 2 = 6 5. R 1 = 2 R 2 = 6 6. R 1 = R 2 = 1 6 correct Explanation: cadena (jc59484) – HW14 – lawn – (55930) 2 When lim n →∞ vextendsingle vextendsingle vextendsingle c n +1 c n vextendsingle vextendsingle vextendsingle = 6 , the Ratio Test ensures that the series ∞ summationdisplay n =0 c n y n is (i) convergent when | y | < 1 6 , and (ii) divergent when | y | > 1 6 . 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 y n- 1 is (i) convergent when | y | < 1 6 , and (ii) divergent when | y | > 1 6 . Consequently, R 1 = R 2 = 1 6 . 003 10.0 points Find a power series representation for the function f ( y ) = 1 y- 2 . 1. f ( y ) =- ∞ summationdisplay n = 0 1 2 n +1 y n correct 2. f ( y ) = ∞ summationdisplay n = 0 1 2 n +1 y n 3. f ( y ) = ∞ summationdisplay n = 0 (- 1) n- 1 2 n +1 y n 4. f ( y ) = ∞ summationdisplay n = 0 (- 1) n 2 n y n 5. f ( y ) =- ∞ summationdisplay n = 0 2 n y n Explanation: We know that 1 1- x = 1 + x + x 2 + . . . = ∞ summationdisplay n =0 x n ....
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