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Unformatted text preview: Version 055 – L EXAM 3 – meth – (54960) 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 whether the sequence { a n } con verges or diverges when a n = ln(3 n 4 ) ln(5 n 3 ) , and if it converges, find the limit. 1. converges with limit = 4 3 correct 2. converges with limit = 0 3. converges with limit = ln 3 ln 5 4. diverges 5. converges with limit = 3 5 Explanation: By properties of logs, ln(3 n 4 ) = ln 3 + 4 ln n , ln(5 n 3 ) = ln 5 + 3 ln n . Thus a n = ln3 + 4 ln n ln5 + 3 ln n = 4 + ln 3 ln n 3 + ln 5 ln n . On the other hand, lim n →∞ ln 3 ln n = lim n →∞ ln 5 ln n = 0 . Properties of limits thus ensure that the given sequence converges with limit = 4 3 . 002 10.0 points Determine if the sequence { a n } converges when a n = 5 + parenleftbigg 10 7 π parenrightbigg n , and if it does, find its limit. 1. limit = 5 + 1 π 2. limit = 5 π 3. limit = 5 1 π 4. the sequence diverges 5. limit = 5 correct Explanation: It is known that x n→ as n → ∞ whenever 1 < x < 1. Now 10 < 7 π , so parenleftbigg 10 7 π parenrightbigg n→ as n → ∞ . Consequently, the given sequence converges and has limit = 5 . 003 10.0 points Determine if the sequence { a n } converges, and if it does, find its limit when a n = 4 n + ( 1) n 6 n + 5 . 1. converges with limit = 2 3 correct 2. sequence does not converge Version 055 – L EXAM 3 – meth – (54960) 2 3. converges with limit = 4 11 4. converges with limit = 1 2 5. converges with limit = 5 6 Explanation: After division by n we see that a n = 4 + ( 1) n n 6 + 5 n . But ( 1) n n , 5 n→ as n → ∞ , so a n → 2 3 as n → ∞ . Conse quently, the sequence converges and has limit = 2 3 . 004 10.0 points If the n th partial sum of an infinite series is S n = 3 n 2 1 5 n 2 + 1 , what is the sum of the series? 1. sum = 3 5 correct 2. sum = 1 3. sum = 1 5 4. series diverges 5. sum = 3 Explanation: By definition sum = lim n →∞ S n = lim n →∞ parenleftBig 3 n 2 1 5 n 2 + 1 parenrightBig . Thus sum = 3 5 . 005 10.0 points Find the values of x for which the series ∞ summationdisplay n = 1 x n 6 n converges, and then find the sum of the series for those values of x . 1. converges: 6 < x < 6 , sum = 6 6 x 2. converges: 6 ≤ x < 6 , sum = 6 6 x 3. converges: 6 ≤ x ≤ 6 , sum = 6 6 x 4. converges: 6 < x ≤ 6 , sum = x 6 x 5. converges: 6 < x < 6 , sum = x 6 x correct 6. converges: 6 ≤ x ≤ 6 , sum = x 6 x Explanation: The given series is a geometric series ∞ summationdisplay n = 1 a r n − 1 in which a = x 6 , r = x 6 . But such a geometric series (i) converges when  r  < 1, with sum a 1 r , and (ii) diverges when  r  ≥ 1....
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This note was uploaded on 02/20/2011 for the course MATH 408L taught by Professor Gogolev during the Fall '09 term at University of Texas at Austin.
 Fall '09
 GOGOLEV

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