EXAM 3 ANSWERS - Sanmiguel Alex Exam 3 Due Dec 4 2007 10:00...

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Sanmiguel, Alex – Exam 3 – Due: Dec 4 2007, 10:00 pm – Inst: E SCHULTZ 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 n + 4 8 n + 7 , and if it does, find its limit. 1. limit = 0 2. limit = 4 7 3. sequence diverges correct 4. limit = ± 1 8 5. limit = 1 8 Explanation: After division, n + 4 8 n + 7 = 1 + 4 n 8 + 7 n . Now 4 n , 7 n 0 as n → ∞ , so lim n → ∞ n + 4 8 n + 7 = 1 8 6 = 0 . Thus as n → ∞ , the values of a n oscillate be- tween values ever closer to ± 1 8 . Consequently, the sequence diverges . keywords: 002 (part 1 of 1) 10 points Determine if the sequence { a n } converges when a n = n 6 n ( n - 7) 6 n , and if it does, find its limit 1. limit = e 7 6 2. limit = e 42 correct 3. sequence diverges 4. limit = e - 42 5. limit = 1 6. limit = e - 7 6 Explanation: By the Laws of Exponents, a n = n - 7 n - 6 n = 1 - 7 n - 6 n = h ‡ 1 - 7 n · n i - 6 . But 1 + x n · n -→ e x as n . Consequently, { a n } converges and has limit = ( e - 7 ) - 6 = e 42 . keywords: sequence, e, exponentials, limit 003 (part 1 of 1) 10 points Determine whether the series 3 + 1 + 1 3 + 1 9 + · · · is convergent or divergent, and if convergent, find its sum. 1. convergent with sum = 9 2 correct
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Sanmiguel, Alex – Exam 3 – Due: Dec 4 2007, 10:00 pm – Inst: E SCHULTZ 2 2. convergent with sum = 1 2 3. convergent with sum = 2 9 4. convergent with sum = 2 5. divergent Explanation: The series 3 + 1 + 1 3 + 1 9 + · · · = X n =1 a r n - 1 is an infinite geometric series in which a = 3 and r = 1 3 . 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 convergent with sum = 9 2 . keywords: 004 (part 1 of 1) 10 points Determine whether the series X n = 0 4 (cos ) 1 3 n is convergent or divergent, and if convergent, find its sum. 1. convergent with sum 3 correct 2. convergent with sum - 6 3. convergent with sum 6 4. divergent 5. convergent with sum - 1 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 4 - 1 3 n = X n = 0 a r n in which a = 4 , 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 . keywords: geometric series, convergent 005 (part 1 of 1) 10 points Determine whether the infinite series X n =1 4( n + 1) 2 n ( n + 2) converges or diverges, and if converges, find its sum. 1. converges with sum 1 4 2. diverges correct 3. converges with sum 1 8
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Sanmiguel, Alex – Exam 3 – Due: Dec 4 2007, 10:00 pm – Inst: E SCHULTZ 3 4. converges with sum 1 6 5. converges with sum 1 2 Explanation: The infinite series X n =1 a n diverges when lim n →∞ a n 6 = 0 .
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