HW11 - ali (fa3364) – HW11 – Radin – (56635) 1 This...

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Unformatted text preview: ali (fa3364) – HW11 – Radin – (56635) 1 This print-out should have 17 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Compute the value of lim n →∞ 4 a n b n 3 a n − 2 b n when lim n →∞ a n = 6 , lim n →∞ b n = − 2 . 1. limit = 25 11 2. limit = − 24 11 correct 3. limit = 24 11 4. limit doesn’t exist 5. limit = − 25 11 Explanation: By properties of limits lim n → 2 4 a n b n = 4 lim n →∞ a n lim n →∞ b n = − 48 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 →∞ 4 a n b n 3 a n − 2 b n = − 24 11 . 002 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 3. a n = parenleftBig − 4 3 parenrightBig n- 1 correct 4. a n = − parenleftBig 4 3 parenrightBig n 5. a n = parenleftBig − 5 4 parenrightBig n- 1 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 003 10.0 points ali (fa3364) – HW11 – Radin – (56635) 2 Determine whether the sequence { a n } con- verges or diverges when a n = 14 n 2 7 n + 2 − 2 n 2 + 3 n + 1 , and if it does, find its limit 1. limit = 5 7 2. limit = 0 3. the sequence diverges 4. limit = 10 7 correct 5. limit = 10 21 Explanation: After bringing the two terms to a common denominator we see that a n = 14 n 3 + 14 n 2 − (7 n + 2) ( 2 n 2 + 3 ) (7 n + 2) ( n + 1) = 10 n 2 − 21 n − 6 7 n 2 + 9 n + 2 . Thus a n = 10 − 21 n − 6 n 2 7 + 9 n + 2 n 2 ....
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HW11 - ali (fa3364) – HW11 – Radin – (56635) 1 This...

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