HW 10 - Portillo, Tom Homework 10 Due: Nov 7 2006, 3:00 am...

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Portillo, Tom – Homework 10 – Due: Nov 7 2006, 3:00 am – Inst: David Benzvi 1 This print-out should have 17 questions. Multiple-choice questions may continue on the next column or page – fnd all choices beFore answering. The due time is Central time. 001 (part 1 oF 1) 10 points Determine iF the sequence { a n } converges when a n = 1 n ln µ 4 5 n + 5 , and iF it does, fnd its limit. 1. limit = ln 4 5 2. the sequence diverges 3. limit = ln 2 5 4. limit = - ln5 5. limit = 0 correct Explanation: AFter division by n we see that 4 5 n + 5 = 4 n 5 + 5 n , so by properties oF logs, a n = 1 n ln 4 n - 1 n ln µ 5 + 5 n . But by known limits (or use L’Hospital), 1 n ln 4 n , 1 n ln µ 5 + 5 n -→ 0 as n → ∞ . Consequently, the sequence { a n } converges and has limit = 0 . keywords: limit, sequence, log Function, 002 (part 1 oF 1) 10 points Determine whether the sequence { a n } con- verges or diverges when a n = 14 n 2 7 n + 1 - 2 n 2 + 3 n + 1 , and iF it does, fnd its limit 1. limit = 4 7 2. limit = 0 3. limit = 6 7 4. the sequence diverges 5. limit = 12 7 correct Explanation: AFter bringing the two terms to a common denominator we see that a n = 14 n 3 + 14 n 2 - (7 n + 1) ( 2 n 2 + 3 ) (7 n + 1)( n + 1) = 12 n 2 - 21 n - 3 7 n 2 + 8 n + 1 . Thus a n = 12 - 21 n - 3 n 2 7 + 8 n + 1 n 2 . But 21 n , 3 n 2 , 8 n , 1 n 2 -→ 0 as n → ∞ . Thus { a n } converges and has limit = 12 7 . keywords: sequence, convergence 003 (part 1 oF 1) 10 points
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Portillo, Tom – Homework 10 – Due: Nov 7 2006, 3:00 am – Inst: David Benzvi 2 Determine if the sequence { a n } converges when a n = n 3 n ( n - 2) 3 n , and if it does, Fnd its limit 1. sequence diverges 2. limit = e 2 3 3. limit = 1 4. limit = e - 6 5. limit = e 6 correct 6. limit = e - 2 3 Explanation: By the Laws of Exponents, a n = µ n - 2 n - 3 n = µ 1 - 2 n - 3 n = h‡ 1 - 2 n · n i - 3 . But 1 + x n · n -→ e x as n → ∞ . Consequently, { a n } converges and has limit = ( e - 2 ) - 3 = e 6 . keywords: sequence, e, exponentials, limit 004 (part 1 of 1) 10 points Determine whether the sequence { a n } con- verges or diverges when a n = ( - 1) n - 1 n n 2 + 3 , and if it converges, Fnd the limit. 1. converges with limit = 0 correct 2. converges with limit = - 1 3 3. sequence diverges 4. converges with limit = 3 5. converges with limit = 1 3 6. converges with limit = - 3 Explanation: After division, a n = ( - 1) n - 1 n n 2 + 3 = ( - 1) n - 1 n + 1 n . Consequently, 0 ≤ | a n | = 1 n + 1 n 1 n . But 1 /n 0 as n → ∞ , so by the Squeeze theorem, lim n →∞ | a n | = 0 . But -| a n | ≤ a n ≤ | a n | , so by the Squeeze theorem again the given sequence { a n } converges and has limit = 0 .
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This note was uploaded on 02/18/2010 for the course MATH 408L taught by Professor Benzvi during the Spring '06 term at North Texas.

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HW 10 - Portillo, Tom Homework 10 Due: Nov 7 2006, 3:00 am...

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