HW 02-solutions

# HW 02-solutions - gonzales(pag757 – HW 02 – Odell...

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Unformatted text preview: gonzales (pag757) – HW 02 – Odell – (57000) 1 This print-out should have 33 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points If lim n →∞ a n = 6 , determine the value, if any, of lim n →∞ a n +9 . 1. limit = − 3 2. limit doesn’t exist 3. limit = 15 4. limit = 6 correct 5. limit = 2 3 Explanation: To say that lim n →∞ a n = 6 means that a n gets as close as we please to 6 for all sufficiently large n . But then a n +9 gets as close as we please to 6 for all sufficiently large n . Consequently, lim n →∞ a n +9 = 6 . 002 10.0 points Determine if the sequence { a n } converges when a n = 1 n ln parenleftbigg 6 4 n + 6 parenrightbigg , and if it does, find its limit. 1. limit = 0 correct 2. the sequence diverges 3. limit = − ln4 4. limit = ln 3 5 5. limit = ln 3 2 Explanation: After division by n we see that 6 4 n + 6 = 6 n 4 + 6 n , so by properties of logs, a n = 1 n ln 6 n − 1 n ln parenleftbigg 4 + 6 n parenrightbigg . But by known limits (or use L’Hospital), 1 n ln 6 n , 1 n ln parenleftbigg 4 + 6 n parenrightbigg −→ as n → ∞ . Consequently, the sequence { a n } converges and has limit = 0 . 003 10.0 points Find a formula for the general term a n of the sequence { a n } ∞ n =1 = braceleftBig 1 , − 4 3 , 16 9 , − 64 27 , . . . bracerightBig , assuming that the pattern of the first few terms continues. 1. a n = − parenleftBig 3 4 parenrightBig n 2. a n = parenleftBig − 5 4 parenrightBig n − 1 3. a n = parenleftBig − 4 3 parenrightBig n − 1 correct 4. a n = − parenleftBig 4 3 parenrightBig n gonzales (pag757) – HW 02 – Odell – (57000) 2 5. a n = − parenleftBig 5 4 parenrightBig n 6. a n = parenleftBig − 3 4 parenrightBig n − 1 Explanation: By inspection, consecutive terms a n − 1 and a n in the sequence { a n } ∞ n =1 = braceleftBig 1 , − 4 3 , 16 9 , − 64 27 , . . . bracerightBig have the property that a n = ra n − 1 = parenleftBig − 4 3 parenrightBig a n − 1 . Thus a n = ra n − 1 = r 2 a n − 2 = . . . = r n − 1 a 1 = parenleftBig − 4 3 parenrightBig n − 1 a 1 . Consequently, a n = parenleftBig − 4 3 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 = 3 n + ( − 1) n 6 n + 2 . 1. converges with limit = 1 2 correct 2. sequence does not converge 3. converges with limit = 3 8 4. converges with limit = 2 3 5. converges with limit = 1 3 Explanation: After division by n we see that a n = 3 + ( − 1) n n 6 + 2 n . But ( − 1) n n , 2 n −→ as n → ∞ , so a n → 1 2 as n → ∞ . Conse- quently, the sequence converges and has limit = 1 2 ....
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HW 02-solutions - gonzales(pag757 – HW 02 – Odell...

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