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exam3solutions - Gaspar Adrian Exam 3 Due Dec 5 2007 1:00...

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Gaspar, Adrian – Exam 3 – Due: Dec 5 2007, 1:00 am – Inst: MC Caputo 1 This print-out should have 18 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. The due time is Central time. 001 (part 1 of 1) 10 points Determine whether the sequence { a n } con- verges or diverges when a n = ( - 1) n 6 n + 5 7 n + 3 , and if it does, find its limit. 1. limit = ± 6 7 2. limit = 0 3. sequence diverges correct 4. limit = 5 3 5. limit = 6 7 Explanation: After division, 6 n + 5 7 n + 3 = 6 + 5 n 7 + 3 n . Now 5 n , 3 n 0 as n → ∞ , so lim n → ∞ 6 n + 5 7 n + 3 = 6 7 6 = 0 . Thus as n → ∞ , the values of a n oscillate be- tween values ever closer to ± 6 7 . Consequently, the sequence diverges . keywords: 002 (part 1 of 1) 10 points Determine if the sequence { a n } converges when a n = n 5 n ( n - 7) 5 n , and if it does, find its limit 1. sequence diverges 2. limit = e 35 correct 3. limit = e 7 5 4. limit = e - 7 5 5. limit = 1 6. limit = e - 35 Explanation: By the Laws of Exponents, a n = n - 7 n - 5 n = 1 - 7 n - 5 n = h ‡ 1 - 7 n · n i - 5 . But 1 + x n · n -→ e x as n . Consequently, { a n } converges and has limit = ( e - 7 ) - 5 = e 35 . keywords: sequence, e, exponentials, limit 003 (part 1 of 1) 10 points Determine whether the series 2 + 3 + 9 2 + 27 4 + · · · is convergent or divergent, and if convergent, find its sum. 1. convergent with sum = 1 4
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Gaspar, Adrian – Exam 3 – Due: Dec 5 2007, 1:00 am – Inst: MC Caputo 2 2. divergent correct 3. convergent with sum = 9 4. convergent with sum = 1 9 5. convergent with sum = 4 Explanation: The series 2 + 3 + 9 2 + 27 4 + · · · = X n =1 a r n - 1 is an infinite geometric series in which a = 2 and r = 3 2 . But such a series is (i) convergent with sum a 1 - r when | r | < 1, (ii) divergent when | r | ≥ 1 . Thus the given series is divergent . keywords: 004 (part 1 of 1) 10 points Determine whether the series X n = 0 2 (cos ) 1 3 n is convergent or divergent, and if convergent, find its sum. 1. convergent with sum - 2 3 2. divergent 3. convergent with sum - 3 2 4. convergent with sum 3 2 correct 5. convergent with sum 3 6. convergent with sum - 3 Explanation: Since cos = ( - 1) n , the given series can be rewritten as an infinite geometric series X n =0 2 - 1 3 n = X n = 0 a r n in which a = 2 , r = - 1 3 . But the series n =0 ar n is (i) convergent with sum a 1 - r when | r | < 1, and (ii) divergent when | r | ≥ 1. Consequently, the given series is convergent with sum 3 2 . keywords: geometric series, convergent 005 (part 1 of 1) 10 points Determine whether the infinite series X n =1 2( n + 1) 2 n ( n + 2) converges or diverges, and if converges, find its sum. 1. converges with sum 1 6 2. converges with sum 1 2 3. converges with sum 1 8
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Gaspar, Adrian – Exam 3 – Due: Dec 5 2007, 1:00 am – Inst: MC Caputo 3 4. diverges correct 5. converges with sum 1 4 Explanation: The infinite series X n =1 a n diverges when lim n →∞ a n 6 = 0 .
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