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Unformatted text preview: zakaria (mmz255) – HW03 – gilbert – (55485) 1 This printout should have 20 questions. Multiplechoice 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 5 n 3 n 3 + 3 is convergent or divergent, and if convergent, find its sum. 1. convergent with sum = 5 2. convergent with sum = 4 5 3. convergent with sum = 1 5 4. divergent correct 5. convergent with sum = 5 4 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 = 5 n 3 n 3 + 3 = 5 1 + 3 n 3 , so lim n →∞ a n = lim n →∞ 5 n 3 n 3 + 3 = 5 negationslash = 0 . Thus the given series is divergent . 002 10.0 points Determine if the series ∞ summationdisplay k = 1 1 + 2 k 3 k converges or diverges, and if it converges, find its sum. 1. converges with sum = 3 2 2. converges with sum = 1 2 3. converges with sum = 1 4. series diverges 5. converges with sum = 2 6. converges with sum = 5 2 correct 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 k = 1 1 3 k = ∞ summationdisplay k = 1 1 3 parenleftBig 1 3 parenrightBig k − 1 is a geometric series with a = r = 1 3 < 1. Thus it converges with sum = 1 2 , while ∞ summationdisplay k = 1 2 k 3 k = ∞ summationdisplay k = 1 2 3 parenleftBig 2 3 parenrightBig k − 1 is a geometric series with a = r = 2 3 < 1. Thus it too converges, and it has sum = 2 . zakaria (mmz255) – HW03 – gilbert – (55485) 2 Consequently, being the sum of two conver gent series, the given series converges with sum = 1 2 + 2 = 5 2 . 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 = tan 2 θ 2. sum = cot 2 θ correct 3. sum = sin 2 θ cos 2 θ 4. sum = sec 2 θ 5. sum = csc 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 f be a continuous, positive, decreasing function on [4 , ∞ ). Compare the values of the integral A = integraldisplay 15 4 f ( x ) dx and the series B = 14 summationdisplay n = 4 f ( n ) . 1. A = B 2. A > B 3. A < B correct Explanation: In the figure 4 5 6 7 8 . . . a 4 a 5 a 6 a 7 the bold line is the graph of f on [4 , ∞ ) and the areas of the rectangles the terms in the series ∞ summationdisplay n = 4 a n , a n = f ( n ) ....
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 Spring '11
 Gilbert
 Mathematical Series, Zakaria

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