h3 - moore(jwm2685 – HW03 – gilbert –(55485 1 This...

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Unformatted text preview: moore (jwm2685) – HW03 – gilbert – (55485) 1 This print-out should have 20 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Determine whether the series ∞ summationdisplay n = 1 2 n 3 4 n 3 + 1 is convergent or divergent, and if convergent, find its sum. 1. convergent with sum = 2 5 2. convergent with sum = 2 3. divergent correct 4. convergent with sum = 1 2 5. convergent with sum = 5 2 Explanation: The infinite series ∞ summationdisplay n =1 a n is divergent when lim n →∞ a n exists but lim n →∞ a n negationslash = 0 . Note for the given series, a n = 2 n 3 4 n 3 + 1 = 2 4 + 1 n 3 , so lim n →∞ a n = lim n →∞ 2 n 3 4 n 3 + 1 = 1 2 negationslash = 0 . Thus the given series is divergent . 002 10.0 points Determine if the series ∞ summationdisplay n = 1 4 + 2 n 5 n converges or diverges, and if it converges, find its sum. 1. converges with sum = 19 12 2. converges with sum = 7 4 3. converges with sum = 17 12 4. converges with sum = 5 3 correct 5. series diverges 6. converges with sum = 3 2 Explanation: An infinite geometric series ∑ ∞ n =1 a r n − 1 (i) converges when | r | < 1 and has sum = a 1 − r , while it (ii) diverges when | r | ≥ 1 . Now ∞ summationdisplay n = 1 4 5 n = ∞ summationdisplay n = 1 4 5 parenleftBig 1 5 parenrightBig n − 1 is a geometric series with a = r = 1 5 < 1. Thus it converges with sum = 1 , while ∞ summationdisplay n = 1 2 n 5 n = ∞ summationdisplay n = 1 2 5 parenleftBig 2 5 parenrightBig n − 1 is a geometric series with a = r = 2 5 < 1. Thus it too converges, and it has sum = 2 3 . moore (jwm2685) – HW03 – gilbert – (55485) 2 Consequently, being the sum of two conver- gent series, the given series converges with sum = 1 + 2 3 = 5 3 . 003 10.0 points Find the sum of the infinite series ∞ summationdisplay k = 1 (cos 2 θ ) k , (0 ≤ θ < 2 π ) , whenever the series converges. 1. sum = sec 2 θ 2. sum = tan 2 θ 3. sum = cot 2 θ correct 4. sum = csc 2 θ 5. sum = sin 2 θ cos 2 θ Explanation: For general θ the series ∞ summationdisplay k = 1 (cos 2 θ ) k is an infinite geometric series with common ratio cos 2 θ . Since the series starts at k = 1, its sum is thus given by cos 2 θ 1 − cos 2 θ = cos 2 θ sin 2 θ . Consequently sum = cot 2 θ . 004 10.0 points Let g be a continuous, positive, decreasing function on [1 , ∞ ). Compare the values of the series A = 13 summationdisplay n = 1 g ( n ) and the integral B = integraldisplay 14 1 g ( z ) dz . 1. A = B 2. A > B correct 3. A < B Explanation: In the figure 1 2 3 4 5 . . . a 1 a 2 a 3 a 4 the bold line is the graph of g on [1 , ∞ ) and the areas of the rectangles the terms in the series ∞ summationdisplay n = 1 a n , a n = g ( n ) ....
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This note was uploaded on 05/02/2011 for the course M 408 D taught by Professor Textbookanswers during the Spring '07 term at University of Texas.

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h3 - moore(jwm2685 – HW03 – gilbert –(55485 1 This...

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