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h4 - moore(jwm2685 – HW04 – gilbert –(55485 1 This...

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Unformatted text preview: moore (jwm2685) – HW04 – gilbert – (55485) 1 This print-out should have 18 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points To apply the root test to an infinite series ∑ n a n the value of ρ = lim n →∞ ( a n ) 1 /n has to be determined. Compute the value of ρ for the series ∞ summationdisplay n =1 3 n + 5 n parenleftbigg 7 4 parenrightbigg n . 1. ρ = 20 7 2. ρ = 35 4 3. ρ = 4 7 4. ρ = 21 4 5. ρ = 7 4 correct Explanation: After division, 3 n + 5 n = 3 parenleftbigg 1 + 5 3 n parenrightbigg , so ( a n ) 1 /n = parenleftbigg 3 braceleftbigg 1 + 5 3 n bracerightbiggparenrightbigg 1 /n 7 4 . But lim n →∞ 3 1 /n parenleftbigg 1 + 5 3 n parenrightbigg 1 /n = 1 as n → ∞ . Consequently, ρ = 7 4 . 002 10.0 points To apply the ratio test to the infinite series summationdisplay n a n , the value λ = lim n →∞ a n +1 a n has to be determined, Compute λ for the series ∞ summationdisplay n =1 1 7 n + 5 sin parenleftbigg 1 n parenrightbigg . 1. λ = 1 correct 2. λ = 0 3. λ = 1 5 4. λ = 1 7 5. λ = 1 12 Explanation: By algebra, sin 1 n +1 sin 1 n = sin 1 n +1 1 n +1 · 1 n sin 1 n · n n + 1 . But lim x → sin x x = 1 , so lim n →∞ sin 1 n +1 1 n +1 = 1 , while lim n →∞ 1 n sin 1 n = 1 also. Thus λ = lim n →∞ n n + 1 bracketleftbigg 7 n + 5 7( n + 1) + 5 bracketrightbigg = lim n →∞ n n + 1 parenleftbigg 7 n + 5 7 n + 7 + 5 parenrightbigg . moore (jwm2685) – HW04 – gilbert – (55485) 2 Consequently, for the given series, λ = 1 . 003 10.0 points Decide whether the series ∞ summationdisplay n =1 3 n parenleftBig n − 2 n parenrightBig n 2 converges or diverges. 1. converges correct 2. diverges Explanation: The given series has the form ∞ summationdisplay n =1 a n , a n = 3 n parenleftBig n − 2 n parenrightBig n 2 . But then | a n | 1 /n = 3 parenleftBig n − 1 n parenrightBig n , in which case lim n → 1 | a n | 1 /n = 3 e 2 < 1 , since lim n →∞ parenleftBig n − 2 n parenrightBig n = lim n →∞ parenleftBig 1 − 2 n parenrightBig n = 1 e 2 < 1 2 2 . Consequently, the Root Test ensures that the given series converges . 004 10.0 points The terms of a series are specified recur- sively by the equations a 1 = 5 , a n +1 = parenleftBig 5 n + 1 4 n + 10 parenrightBig a n . Determine whether the series ∞ summationdisplay n =1 a n converges or diverges. 1. converges 2. neither converges nor diverges 3. diverges correct Explanation: From the recursive formula we see that lim n →∞ a n +1 a n = lim n →∞ 5 n + 1 4 n + 10 = 5 4 . Consequently, by the Ratio Test the series diverges. 005 10.0 points Decide whether the series ∞ summationdisplay n =1 (2 n )!...
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