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Gilbert_Hmwk04sol

# Gilbert_Hmwk04sol - moseley(cmm3869 HW04 Gilbert(56380 This...

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moseley (cmm3869) – HW04 – Gilbert – (56380) 1 This print-out should have 24 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 =0 4 (3) n is convergent or divergent, and if convergent, find its sum. 1. convergent, sum = 5 2 2. convergent, sum = 5 2 3. convergent, sum = 1 4. divergent correct 5. convergent, sum = 2 Explanation: The given series is an infinite geometric series summationdisplay n =0 a r n with a = 4 and r = 3. 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 = cot 2 θ correct 2. sum = csc 2 θ 3. sum = tan 2 θ 4. sum = sec 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 . 003 10.0 points Determine whether the series summationdisplay n =1 n 3 5 n 3 + 3 is convergent or divergent, and if convergent, find its sum. 1. convergent with sum = 8 2. convergent with sum = 1 5 3. convergent with sum = 1 8 4. convergent with sum = 5 5. divergent correct

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moseley (cmm3869) – 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 = n 3 5 n 3 + 3 = 1 5 + 3 n 3 , so lim n →∞ a n = lim n →∞ n 3 5 n 3 + 3 = 1 5 negationslash = 0 . Thus the given series is divergent . 004 10.0 points Determine if the series summationdisplay n =1 2 + 3 n 4 n converges or diverges, and if it converges, find its sum. 1. converges with sum = 3 2. converges with sum = 8 3 3. series diverges 4. converges with sum = 7 3 5. converges with sum = 11 3 correct 6. converges with sum = 10 3 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 2 4 n = summationdisplay n =1 1 2 parenleftBig 1 4 parenrightBig n 1 is a geometric series with a = r = 1 4 < 1. Thus it converges with sum = 2 3 , while summationdisplay n =1 3 n 4 n = summationdisplay n =1 3 4 parenleftBig 3 4 parenrightBig n 1 is a geometric series with a = r = 3 4 < 1. Thus it too converges, and it has sum = 3 . Consequently, being the sum of two conver- gent series, the given series converges with sum = 2 3 + 3 = 11 3 . 005 10.0 points Determine whether the infinite series summationdisplay n =1 tan 1 parenleftBig 3 n 2 4 n + 1 parenrightBig converges or diverges, and if it converges, find its sum.
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Gilbert_Hmwk04sol - moseley(cmm3869 HW04 Gilbert(56380 This...

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