This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: padilla (tp5647) Exam3Review Gilbert (56650) 1 This printout should have 29 questions. Multiplechoice 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 ....
View
Full
Document
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 TexasTyler.
 Spring '09
 GILBERT

Click to edit the document details