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Unformatted text preview: Villarreal (nav277) – Review 3 – Odell – (58340) 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 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 = 3 n n + 1 . 1. a n = 1 2 n 2 2. a n = 3 2 n 3. a n = 1 2 n 4. a n = 1 n ( n + 1) 5. a n = 3 2 n 2 6. a n = 3 n ( n + 1) correct 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 = 3 n n + 1 = 3 − 3 n + 1 . Thus a 1 = 3 2 , while a n = 3 n − 3 n + 1 , ( n > 1) . Consequently, a n = 3 n − 3 n + 1 = 3 n ( n + 1) for all n . 002 10.0 points Let f be a continuous, positive, decreasing function on [4 , ∞ ). Compare the values of the series A = 16 summationdisplay n = 4 f ( n ) and the integral B = integraldisplay 17 4 f ( z ) dz . 1. A > B correct 2. A = B 3. A < B Explanation: In the figure 4 5 6 7 8 . . . a 4 a 5 a 6 a 7 the bold line is the graph of f on [4 , ∞ ) and the areas of the rectangles the terms in the series ∞ summationdisplay n = 4 a n , a n = f ( n ) . Clearly from this figure we see that f (4) > integraldisplay 5 4 f ( z ) dz, f (5) > integraldisplay 6 5 f ( z ) dz , Villarreal (nav277) – Review 3 – Odell – (58340) 2 while f (6) > integraldisplay 7 6 f ( z ) dz, f (7) > integraldisplay 8 7 f ( z ) dz , and so on. Consequently, A > B . keywords: 003 10.0 points To apply the root test to an infinite series ∑ k a k , the value of ρ = lim k →∞  a k  1 /k has to be determined. Compute the value of ρ for the series ∞ summationdisplay k = 1 3 k k (ln k + 8) k . 1. ρ = ∞ 2. ρ = 3 3. ρ = 24 4. ρ = 8 5. ρ = 0 correct Explanation: For the given series ( a k ) 1 /k = 3 1 /k parenleftbigg ln k + 8 k parenrightbigg = 3 1 /k parenleftbigg ln k k + 8 k parenrightbigg . But 3 1 /k −→ 1 , ln k k −→ as k → ∞ . Consequently, ρ = 0 . 004 10.0 points Determine whether the series ∞ summationdisplay n =0 parenleftbigg 4 3 parenrightbigg n/ 2 is convergent or divergent, and if convergent, find its sum. 1. convergent with sum = √ 3 √ 3 − 2 2. convergent with sum = 2 √ 3 − 2 3. convergent with sum = √ 3 − 2 2 4. convergent with sum = √ 3 − 2 √ 3 5. divergent correct Explanation: The infinite series ∞ summationdisplay n =0 parenleftbigg 4 3 parenrightbigg n/ 2 is an infinite geometric series ∑ ∞ n =0 ar n with a = 1 and r = 2 / √ 3. But ∑ ∞ n =0 ar n is (i) convergent with sum a 1 − r when  r  < 1, and (ii) divergent when  r  ≥ 1 . So the given series is divergent . 005 10.0 points Decide which, if any, of the following series converge. Villarreal (nav277) – Review 3 – Odell – (58340) 3 ( A ) ∞ summationdisplay n = 1 n 6 n + 3 parenleftbigg 1 2 parenrightbigg n ( B ) ∞ summationdisplay n = 1 parenleftbigg 6 n + 7 n 3 + 6 parenrightbigg n 1. neither of them...
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 Fall '08
 Cepparo
 Mathematical Series, lim, Villarreal

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