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Unformatted text preview: Massaro, Michael Exam 3 Due: Dec 5 2007, 1:00 am Inst: Shinko Harper 1 This printout should have 18 questions. Multiplechoice 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 3 n + 4 7 n + 3 , and if it does, find its limit. 1. limit = 3 7 2. limit = 3 7 3. limit = 0 4. sequence diverges correct 5. limit = 4 3 Explanation: After division, 3 n + 4 7 n + 3 = 3 + 4 n 7 + 3 n . Now 4 n , 3 n 0 as n , so lim n 3 n + 4 7 n + 3 = 3 7 6 = 0 . Thus as n , the values of a n oscillate be tween values ever closer to 3 7 . Consequently, the sequence diverges . keywords: 002 (part 1 of 1) 10 points Determine if the sequence { a n } converges when a n = n 4 n ( n 8) 4 n , and if it does, find its limit 1. limit = e 32 2. limit = e 32 correct 3. limit = e 2 4. sequence diverges 5. limit = e 2 6. limit = 1 Explanation: By the Laws of Exponents, a n = n 8 n  4 n = 1 8 n  4 n = h 1 8 n n i 4 . But 1 + x n n e x as n . Consequently, { a n } converges and has limit = ( e 8 ) 4 = e 32 . 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 9 Massaro, Michael Exam 3 Due: Dec 5 2007, 1:00 am Inst: Shinko Harper 2 2. convergent with sum = 4 3. convergent with sum = 1 4 4. divergent correct 5. convergent with sum = 9 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 n ) 1 2 n is convergent or divergent, and if convergent, find its sum. 1. convergent with sum 4 2. divergent 3. convergent with sum 4 4. convergent with sum 3 4 5. convergent with sum 4 3 6. convergent with sum 4 3 correct Explanation: Since cos n = ( 1) n , the given series can be rewritten as an infinite geometric series X n =0 2  1 2 n = X n = 0 a r n in which a = 2 , r = 1 2 . 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 4 3 . keywords: geometric series, convergent 005 (part 1 of 1) 10 points Determine whether the infinite series X n =1 3( n + 1) 2 n ( n + 2) converges or diverges, and if converges, find its sum....
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 Spring '08
 RAdin

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