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Unformatted text preview: rhodes (ajr2283) – HW04 – Gilbert – (56380) 1 This printout should have 24 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 =0 3 (2) n is convergent or divergent, and if convergent, find its sum. 1. divergent correct 2. convergent, sum = 1 3. convergent, sum = − 4 4. convergent, sum = − 3 5. convergent, sum = 4 Explanation: The given series is an infinite geometric series ∞ summationdisplay n = 0 a r n with a = 3 and r = 2. But the sum of such a series is (i) convergent with sum a 1 − r when  r  < 1, (ii) divergent when  r  ≥ 1. Consequently, the given series is divergent . 002 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 = sin 2 θ cos 2 θ 3. sum = cot 2 θ correct 4. sum = tan 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 . 003 10.0 points Determine whether the series ∞ summationdisplay n =1 4 n n + 5 is convergent or divergent, and if convergent, find its sum. 1. convergent with sum = 1 4 2. convergent with sum = 2 3 3. convergent with sum = 4 4. divergent correct 5. convergent with sum = 3 2 rhodes (ajr2283) – HW04 – Gilbert – (56380) 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 = 4 n n + 5 = 4 1 + 5 n , so lim n →∞ a n = lim n →∞ 4 n n + 5 = 4 negationslash = 0 . Thus the given series is divergent . 004 10.0 points Determine if the series ∞ summationdisplay m = 1 4 + 3 m 5 m converges or diverges, and if it converges, find its sum. 1. converges with sum = 21 8 2. converges with sum = 19 8 3. converges with sum = 5 2 correct 4. converges with sum = 11 4 5. series diverges 6. converges with sum = 9 4 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 m = 1 4 5 m = ∞ summationdisplay m = 1 4 5 parenleftBig 1 5 parenrightBig m − 1 is a geometric series with a = r = 1 5 < 1. Thus it converges with sum = 1 , while ∞ summationdisplay m =1 3 m 5 m = ∞ summationdisplay m =1 3 5 parenleftBig 3 5 parenrightBig m − 1 is a geometric series with a = r = 3 5 < 1....
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 Spring '07
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 Multivariable Calculus, Mathematical Series, lim, Rhodes

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