M408L_HW11

# M408L_HW11 - hyun(hh7953 – HW11 – gogolev –(57440 1...

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Unformatted text preview: hyun (hh7953) – HW11 – gogolev – (57440) 1 This print-out should have 19 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Determine if the sequence { a n } converges when a n = 1 n ln parenleftbigg 5 6 n + 5 parenrightbigg , and if it does, find its limit. 1. limit = ln 5 6 2. limit = 0 correct 3. limit = − ln 6 4. the sequence diverges 5. limit = ln 5 11 Explanation: After division by n we see that 5 6 n + 5 = 5 n 6 + 5 n , so by properties of logs, a n = 1 n ln 5 n − 1 n ln parenleftbigg 6 + 5 n parenrightbigg . But by known limits (or use L’Hospital), 1 n ln 5 n , 1 n ln parenleftbigg 6 + 5 n parenrightbigg −→ as n → ∞ . Consequently, the sequence { a n } converges and has limit = 0 . 002 10.0 points Find a formula for the general term a n of the sequence { a n } ∞ n =1 = braceleftBig 3 , 8 , 13 , 18 , . . . bracerightBig , assuming that the pattern of the first few terms continues. 1. a n = 6 n − 3 2. a n = 4 n − 1 3. a n = n + 4 4. a n = n + 5 5. a n = 5 n − 2 correct Explanation: By inspection, consecutive terms a n − 1 and a n in the sequence { a n } ∞ n =1 = braceleftBig 3 , 8 , 13 , 18 , . . . bracerightBig have the property that a n − a n − 1 = d = 5 . Thus a n = a n − 1 + d = a n − 2 + 2 d = . . . = a 1 + ( n − 1) d = 3 + 5( n − 1) . Consequently, a n = 5 n − 2 . keywords: 003 10.0 points Find a formula for the general term a n of the sequence { a n } ∞ n =1 = braceleftBig 1 , − 3 4 , 9 16 , − 27 64 , . . . bracerightBig , assuming that the pattern of the first few terms continues. 1. a n = parenleftBig − 4 3 parenrightBig n − 1 hyun (hh7953) – HW11 – gogolev – (57440) 2 2. a n = parenleftBig − 4 5 parenrightBig n − 1 3. a n = − parenleftBig 4 5 parenrightBig n 4. a n = parenleftBig − 3 4 parenrightBig n − 1 correct 5. a n = − parenleftBig 4 3 parenrightBig n 6. a n = − parenleftBig 3 4 parenrightBig n Explanation: By inspection, consecutive terms a n − 1 and a n in the sequence { a n } ∞ n =1 = braceleftBig 1 , − 3 4 , 9 16 , − 27 64 , . . . bracerightBig have the property that a n = ra n − 1 = parenleftBig − 3 4 parenrightBig a n − 1 . Thus a n = ra n − 1 = r 2 a n − 2 = . . . = r n − 1 a 1 = parenleftBig − 3 4 parenrightBig n − 1 a 1 . Consequently, a n = parenleftBig − 3 4 parenrightBig n − 1 since a 1 = 1. keywords: sequence, common ratio 004 10.0 points Determine if the sequence { a n } converges, and if it does, find its limit when a n = 6 n 5 − 3 n 3 + 1 8 n 4 + n 2 + 1 ....
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## This note was uploaded on 01/19/2010 for the course M 57440 taught by Professor Gogolev during the Fall '09 term at University of Texas.

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M408L_HW11 - hyun(hh7953 – HW11 – gogolev –(57440 1...

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