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Unformatted text preview: Panjwani, Sameer Exam 3 Due: Dec 5 2007, 1:00 am Inst: James Rath 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 2 n + 4 n + 3 , and if it does, find its limit. 1. limit = 0 2. sequence diverges correct 3. limit = 2 4. limit = 2 5. limit = 4 3 Explanation: After division, 2 n + 4 n + 3 = 2 + 4 n 1 + 3 n . Now 4 n , 3 n 0 as n , so lim n 2 n + 4 n + 3 = 2 6 = 0 . Thus as n , the values of a n oscillate be tween values ever closer to 2. Consequently, the sequence diverges . keywords: 002 (part 1 of 1) 10 points Determine if the sequence { a n } converges when a n = n 2 n ( n 6) 2 n , and if it does, find its limit 1. limit = e 12 correct 2. sequence diverges 3. limit = e 3 4. limit = e 3 5. limit = e 12 6. limit = 1 Explanation: By the Laws of Exponents, a n = n 6 n  2 n = 1 6 n  2 n = h 1 6 n n i 2 . But 1 + x n n e x as n . Consequently, { a n } converges and has limit = ( e 6 ) 2 = e 12 . keywords: sequence, e, exponentials, limit 003 (part 1 of 1) 10 points Determine whether the series 3 + 2 + 4 3 + 8 9 + is convergent or divergent, and if convergent, find its sum. 1. convergent with sum = 1 9 Panjwani, Sameer Exam 3 Due: Dec 5 2007, 1:00 am Inst: James Rath 2 2. divergent 3. convergent with sum = 4 4. convergent with sum = 1 4 5. convergent with sum = 9 correct Explanation: The series 3 + 2 + 4 3 + 8 9 + = X n =1 a r n 1 is an infinite geometric series in which a = 3 and r = 2 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 . keywords: 004 (part 1 of 1) 10 points Determine whether the series X n = 0 3 (cos n ) 3 4 n is convergent or divergent, and if convergent, find its sum. 1. convergent with sum 12 2. convergent with sum 12 3. convergent with sum 7 12 4. convergent with sum 12 7 5. convergent with sum 12 7 correct 6. divergent Explanation: Since cos n = ( 1) n , the given series can be rewritten as an infinite geometric series X n =0 3  3 4 n = X n = 0 a r n in which a = 3 , r = 3 4 . 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 12 7 . keywords: geometric series, convergent 005 (part 1 of 1) 10 points Determine whether the infinite series X n =1 ( n + 1) 2 n ( n + 2) converges or diverges, and if converges, find its sum....
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This note was uploaded on 03/18/2008 for the course M 408L taught by Professor Radin during the Fall '08 term at University of Texas at Austin.
 Fall '08
 RAdin
 Calculus

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