review 3 - padilla (tp5647) Exam3Review Gilbert (56650) 1...

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Unformatted text preview: padilla (tp5647) Exam3Review Gilbert (56650) 1 This print-out should have 29 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. 001 0.0 points Determine if the sequence { a n } converges when a n = 1 n ln parenleftbigg 5 4 n + 4 parenrightbigg , and if it does, find its limit. 1. limit = 0 correct 2. limit = ln 5 8 3. the sequence diverges 4. limit = ln 4 5. limit = ln 5 4 Explanation: After division by n we see that 5 4 n + 4 = 5 n 4 + 4 n , so by properties of logs, a n = 1 n ln 5 n 1 n ln parenleftbigg 4 + 4 n parenrightbigg . But by known limits (or use LHospital), 1 n ln 5 n , 1 n ln parenleftbigg 4 + 4 n parenrightbigg as n . Consequently, the sequence { a n } converges and has limit = 0 . 002 0.0 points Determine if the sequence { a n } converges when a n = n 3 n ( n 2) 3 n , and if it does, find its limit 1. limit = e 2 3 2. limit = e 2 3 3. sequence diverges 4. limit = e 6 correct 5. limit = 1 6. limit = e 6 Explanation: By the Laws of Exponents, a n = parenleftbigg n 2 n parenrightbigg 3 n = parenleftbigg 1 2 n parenrightbigg 3 n = bracketleftBigparenleftBig 1 2 n parenrightBig n bracketrightBig 3 . But parenleftBig 1 + x n parenrightBig n e x as n . Consequently, { a n } converges and has limit = ( e 2 ) 3 = e 6 . 003 0.0 points Determine if the limit lim n n ( n + 8 n 6) exists, and find its value when it does. 1. limit = 1 2. limit = 7 correct 3. limit doesnt exist 4. limit = 2 padilla (tp5647) Exam3Review Gilbert (56650) 2 5. limit = 14 Explanation: By rationalization, n + 8 n 6 = ( n + 8) ( n 6) n + 8 + n 6 = 14 n + 8 + n 6 . On the other hand, n n + 8 + n 6 = 1 radicalbigg 1 + 8 n + radicalbigg 1 6 n . Since lim n radicalbigg 1 + 8 n = lim n radicalbigg 1 6 n = 1 , it thus follows by properties of limits that lim n radicalbigg 1 + 8 n + radicalbigg 1 6 n exists and has value 2. Consequently, again by properties of limits, the limit lim n n ( n + 8 n 6) exists and limit = 7 . 004 0.0 points Determine whether the series summationdisplay n =0 2 (cos n ) parenleftbigg 3 4 parenrightbigg n is convergent or divergent, and if convergent, find its sum. 1. divergent 2. convergent with sum 8 7 3. convergent with sum 7 8 4. convergent with sum 8 5. convergent with sum 8 7 correct 6. convergent with sum 8 Explanation: Since cos n = ( 1) n , the given series can be rewritten as an infinite geometric series summationdisplay n =0 2 parenleftbigg 3 4 parenrightbigg n = summationdisplay n = 0 a r n in which a = 2 , r = 3 4 ....
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This note was uploaded on 09/09/2009 for the course M 408L taught by Professor Gilbert during the Spring '09 term at University of Texas-Tyler.

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review 3 - padilla (tp5647) Exam3Review Gilbert (56650) 1...

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