Practice Exam 3-solutions

# Practice Exam 3-solutions - sl7433 – Practice Exam 3 –...

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Unformatted text preview: sl7433 – Practice Exam 3 – Radin – (58415) 1 This print-out should have 23 questions. Multiple-choice 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 = ( − 1) n parenleftbigg n + 4 7 n + 4 parenrightbigg , and if it does, find its limit. 1. sequence diverges correct 2. limit = 1 7 3. limit = 1 4. limit = 0 5. limit = ± 1 7 Explanation: After division, n + 4 7 n + 4 = 1 + 4 n 7 + 4 n . Now 4 n , 4 n → 0 as n → ∞ , so lim n →∞ n + 4 7 n + 4 = 1 7 negationslash = 0 . Thus as n → ∞ , the values of a n oscillate be- tween values ever closer to ± 1 7 . Consequently, the sequence diverges . 002 10.0 points Determine whether the series 4 + 3 + 9 4 + 27 16 + ··· is convergent or divergent, and if convergent, find its sum. 1. convergent with sum = 1 16 2. convergent with sum = 9 3. convergent with sum = 16 correct 4. convergent with sum = 1 9 5. divergent Explanation: The series 4 + 3 + 9 4 + 27 16 + ··· = ∞ summationdisplay n =1 a r n − 1 is an infinite geometric series in which a = 4 and r = 3 4 . But such a series is (i) convergent with sum a 1 − r when | r | < 1, (ii) divergent when | r | ≥ 1 . Thus the given series is convergent with sum = 16 . 003 10.0 points Determine whether the infinite series ∞ summationdisplay n =1 3( n + 1) 2 n ( n + 2) converges or diverges, and if converges, find its sum. 1. converges with sum = 3 2 2. diverges correct 3. converges with sum = 3 4 sl7433 – Practice Exam 3 – Radin – (58415) 2 4. converges with sum = 3 5. converges with sum = 3 8 Explanation: By the Divergence Test, an infinite series ∑ n a n diverges when lim n →∞ a n negationslash = 0 . Now, for the given series, a n = 3( n + 1) 2 n ( n + 2) = 3 n 2 + 6 n + 3 n 2 + 2 n . But then, lim n →∞ a n = 3 negationslash = 0 . Consequently, the Divergence Test says that the given series diverges . keywords: infinite series, Divergence Test, ra- tional function 004 10.0 points To apply the root test to an infinite series ∑ n a n the value of ρ = lim n →∞ | a n | 1 /n has to be determined. Compute the value of ρ for the series ∞ summationdisplay n = 1 parenleftbigg 3 n + 4 5 n parenrightbigg 2 n . 1. ρ = 16 25 2. ρ = 9 25 correct 3. ρ = 3 5 4. ρ = 16 9 5. ρ = 4 5 Explanation: After division, 3 n + 4 5 n = 3 + 4 /n 5 , so | a n | 1 /n = parenleftBig 3 + 4 /n 5 parenrightBig 2 . On the other hand, lim n →∞ 3 + 4 /n 5 = 3 5 . Consequently, ρ = 9 25 . keywords: /* If you use any of these, fix the comment symbols. 005 10.0 points Determine whether the series ∞ summationdisplay n =1 ( − 1) n − 1 cos parenleftBig 1 5 n parenrightBig is absolutely convergent, conditionally con- vergent or divergent....
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## This note was uploaded on 03/29/2009 for the course M 408L taught by Professor Radin during the Spring '08 term at University of Texas.

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Practice Exam 3-solutions - sl7433 – Practice Exam 3 –...

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