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Unformatted text preview: hussain (tsh476) – HW 02 – rusin – (55565) 1 This printout should have 20 questions. Multiplechoice 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 + 2 parenrightbigg , and if it does, find its limit. 1. limit = − ln 6 2. limit = ln 5 6 3. limit = 0 correct 4. limit = ln 5 8 5. the sequence diverges Explanation: After division by n we see that 5 6 n + 2 = 5 n 6 + 2 n , so by properties of logs, a n = 1 n ln 5 n − 1 n ln parenleftbigg 6 + 2 n parenrightbigg . But by known limits (or use L’Hospital), 1 n ln 5 n , 1 n ln parenleftbigg 6 + 2 n parenrightbigg −→ as n → ∞ . Consequently, the sequence { a n } converges and has limit = 0 . 002 10.0 points Determine if { a n } converges when a n = 4 √ n 3 − 3 √ n 2 , and if it does, find its limit. 1. limit = 4 2. limit = − 1 6 3. limit = 3 4. sequence diverges correct 5. limit = 1 Explanation: After simplification, a n = 4 n √ n − 3 n = n ( 4 √ n − 3 ) , so a n → ∞ as n → ∞ . But then the sequence is unbounded, and hence diverges . 003 10.0 points Determine if the sequence { a n } converges, and if it does, find its limit when a n = parenleftBig n − 1 n + 1 parenrightBig − 2 n . 1. limit = e − 2 2. limit = e 4 correct 3. limit = e 2 4. limit = e − 4 5. limit = 1 6. does not converge Explanation: By the Laws of Exponents, a n = parenleftBig n + 1 n − 1 parenrightBig 2 n = parenleftBig 1 + 1 n parenrightBig 2 n parenleftBig 1 − 1 n parenrightBig 2 n . hussain (tsh476) – HW 02 – rusin – (55565) 2 But lim n →∞ parenleftBig 1 + x n parenrightBig n = e x . Consequently, by Properties of Limits the given limit exists and limit = e 4 . 004 10.0 points Determine whether the sequence { a n } con verges or diverges when a n = ( − 1) n − 1 n n 2 + 4 , and if it converges, find the limit. 1. converges with limit = − 1 4 2. converges with limit = 1 4 3. converges with limit = − 4 4. sequence diverges 5. converges with limit = 0 correct 6. converges with limit = 4 Explanation: After division, a n = ( − 1) n − 1 n n 2 + 4 = ( − 1) n − 1 n + 4 n . Consequently, ≤  a n  = 1 n + 4 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 . keywords: 005 10.0 points Determine if the sequence { a n } converges when a n = (2 n + 1)! (2 n − 1)! , and if it converges, find the limit. 1. converges with limit = 1 4 2. converges with limit = 4 3. converges with limit = 0 4. converges with limit = 1 5. does not converge correct Explanation: By definition, m ! is the product m ! = 1 . 2 . 3 . . . . .m of the first m positive integers. When m = 2 n − 1, therefore, (2 n − 1)! = 1 . 2 . 3 . . . . (2 n − 1) , while (2 n + 1)! = 1 . 2 . 3 . . . . .....
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This note was uploaded on 04/03/2011 for the course MATH 408D taught by Professor Chu during the Spring '09 term at University of Texas.
 Spring '09
 Chu

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