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Unformatted text preview: Rehman (aar638) HW11 sachse (56620) 1 This printout should have 17 questions. Multiplechoice questions may continue on the next column or page find all choices before answering. 001 10.0 points Compute the value of lim n 2 a n b n 3 a n 2 b n when lim n a n = 6 , lim n b n = 2 . 1. limit = 12 11 2. limit = 13 11 3. limit = 12 11 correct 4. limit doesnt exist 5. limit = 13 11 Explanation: By properties of limits lim n 2 2 a n b n = 2 lim n a n lim n b n = 24 while lim n (3 a n 2 b n ) = 3 lim n a n 2 lim n b n = 22 negationslash = 0 . Thus, by properties of limits again, lim n 2 a n b n 3 a n 2 b n = 12 11 . 002 10.0 points Find a formula for the general term a n of the sequence { a n } n =1 = braceleftBig 1 , 5 2 , 25 4 , 125 8 , . . . bracerightBig , assuming that the pattern of the first few terms continues. 1. a n = parenleftBig 5 2 parenrightBig n 2. a n = parenleftBig 5 2 parenrightBig n 1 correct 3. a n = parenleftBig 2 parenrightBig n 4. a n = parenleftBig 2 5 parenrightBig n 5. a n = parenleftBig 2 parenrightBig n 1 6. a n = parenleftBig 2 5 parenrightBig n 1 Explanation: By inspection, consecutive terms a n 1 and a n in the sequence { a n } n =1 = braceleftBig 1 , 5 2 , 25 4 , 125 8 , . . . bracerightBig have the property that a n = ra n 1 = parenleftBig 5 2 parenrightBig a n 1 . Thus a n = ra n 1 = r 2 a n 2 = . . . = r n 1 a 1 = parenleftBig 5 2 parenrightBig n 1 a 1 . Consequently, a n = parenleftBig 5 2 parenrightBig n 1 since a 1 = 1. keywords: sequence, common ratio 003 10.0 points Rehman (aar638) HW11 sachse (56620) 2 Determine whether the sequence { a n } con verges or diverges when a n = 10 n 2 5 n + 4 2 n 2 + 5 n + 1 , and if it does, find its limit 1. limit = 0 2. limit = 1 5 3. the sequence diverges 4. limit = 2 5 correct 5. limit = 2 15 Explanation: After bringing the two terms to a common denominator we see that a n = 10 n 3 + 10 n 2 (5 n + 4) ( 2 n 2 + 5 ) (5 n + 4) ( n + 1) = 2 n 2 25 n 20 5 n 2 + 9 n + 4 . Thus a n = 2 25 n 20 n 2 5 + 9 n + 4 n 2 ....
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This note was uploaded on 02/01/2011 for the course MATH 408L taught by Professor Gogolev during the Fall '09 term at University of Texas at Austin.
 Fall '09
 GOGOLEV
 Calculus

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