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Unformatted text preview: Cheung, Anthony – Homework 10 – Due: Nov 7 2006, 3:00 am – Inst: David Benzvi 1 This printout should have 17 questions. Multiplechoice questions may continue on the next column or page – find 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 3 5 n + 1 ¶ , and if it does, find its limit. 1. limit = ln 1 2 2. limit = ln 3 5 3. limit = ln5 4. the sequence diverges 5. limit = 0 correct Explanation: After division by n we see that 3 5 n + 1 = 3 n 5 + 1 n , so by properties of logs, a n = 1 n ln 3 n 1 n ln µ 5 + 1 n ¶ . But by known limits (or use L’Hospital), 1 n ln 3 n , 1 n ln µ 5 + 1 n ¶→ 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 = 12 n 2 6 n + 3 2 n 2 + 6 n + 1 , and if it does, find its limit 1. limit = 0 2. limit = 1 3 3. limit = 1 correct 4. limit = 1 2 5. the sequence diverges Explanation: After bringing the two terms to a common denominator we see that a n = 12 n 3 + 12 n 2 (6 n + 3) ( 2 n 2 + 6 ) (6 n + 3)( n + 1) = 6 n 2 36 n 18 6 n 2 + 9 n + 3 . Thus a n = 6 36 n 18 n 2 6 + 9 n + 3 n 2 . But 36 n , 18 n 2 , 9 n , 3 n 2→ as n → ∞ . Thus { a n } converges and has limit = 1 . keywords: sequence, convergence 003 (part 1 of 1) 10 points Determine if the sequence { a n } converges when a n = n 4 n ( n 1) 4 n , Cheung, Anthony – Homework 10 – Due: Nov 7 2006, 3:00 am – Inst: David Benzvi 2 and if it does, find its limit 1. sequence diverges 2. limit = e 4 correct 3. limit = e 1 4 4. limit = e 1 4 5. limit = e 4 6. limit = 1 Explanation: By the Laws of Exponents, a n = µ n 1 n ¶ 4 n = µ 1 1 n ¶ 4 n = h‡ 1 1 n · n i 4 . But ‡ 1 + x n · n→ e x as n → ∞ . Consequently, { a n } converges and has limit = ( e 1 ) 4 = e 4 . 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 + 9 , and if it converges, find the limit. 1. sequence diverges 2. converges with limit = 9 3. converges with limit = 1 9 4. converges with limit = 0 correct 5. converges with limit = 1 9 6. converges with limit = 9 Explanation: After division, a n = ( 1) n 1 n n 2 + 9 = ( 1) n 1 n + 1 n ....
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 Spring '08
 RAdin
 Calculus, Limit, David Benzvi

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