Exam 1-solutions

# Exam 1-solutions - Version 034 – Exam 1 – mann...

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Unformatted text preview: Version 034 – Exam 1 – mann – (54675) 1 This print-out should have 10 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Find the n th term, a n , of an infinite series ∑ ∞ n =1 a n when the n th partial sum, S n , of the series is given by S n = 2 n n + 1 . 1. a n = 5 2 n 2. a n = 5 2 n 2 3. a n = 1 n 2 4. a n = 2 n ( n + 1) correct 5. a n = 1 n 6. a n = 5 n ( n + 1) Explanation: Since S n = a 1 + a 2 + ··· + a n , we see that a 1 = S 1 , a n = S n- S n- 1 ( n > 1) . But S n = 2 n n + 1 = 2- 2 n + 1 . Thus a 1 = 1, while a n = 2 n- 2 n + 1 , ( n > 1) . Consequently, a n = 2 n- 2 n + 1 = 2 n ( n + 1) for all n . 002 10.0 points Determine whether the series ∞ summationdisplay n =0 1 √ n + 4 cos nπ is conditionally convergent, absolutely con- vergent or divergent. 1. divergent 2. absolutely convergent 3. conditionally convergent correct Explanation: Since cos nπ = (- 1) n , the given series can be rewritten as the alternating series ∞ summationdisplay n =0 (- 1) n 1 √ n + 4 = ∞ summationdisplay n = 0 (- 1) n f ( n ) with f ( x ) = 1 √ x + 4 . Now f ( n ) = 1 √ n + 4 > 1 √ n + 5 = f ( n + 1) for all n , while lim n →∞ f ( n ) = lim n →∞ 1 √ n + 4 = 0 . Consequently, by the Alternating Series Test, the given series converges. On the other hand, by the Limit Comparison Test and the p-series test with p = 1 / 2, we see that the series ∞ summationdisplay n =0 f ( n ) is divergent. Consequently, the given series is conditionally convergent . 003 10.0 points Version 034 – Exam 1 – mann – (54675) 2 Which one of the following properties does the series ∞ summationdisplay n =1 (- 1) n- 1 n 2 + 5 3 n have?...
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## This note was uploaded on 11/17/2010 for the course PHY 56630 taught by Professor Coker during the Spring '10 term at University of Texas.

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Exam 1-solutions - Version 034 – Exam 1 – mann...

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