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# review 3 - padilla(tp5647 Exam3Review Gilbert(56650 This...

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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 L’Hospital), 1 n ln 5 n , 1 n ln parenleftbigg 4 + 4 n parenrightbigg −→ 0 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 = bracketleftBig parenleftBig 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 doesn’t exist 4. limit = 2

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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 ) 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 = ( 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 . 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 8 7 .
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review 3 - padilla(tp5647 Exam3Review Gilbert(56650 This...

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