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FinalCalcReview3 - Villarreal(nav277 Review 3 Odell(58340...

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Villarreal (nav277) – Review 3 – Odell – (58340) 1 This print-out should have 20 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 = 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 ,
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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 −→ 0 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.
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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 2. B only 3. both of them correct 4. A only Explanation: We compute one of lim n → ∞ a n +1 a n , lim n → ∞ ( a n ) 1 /n for each of the given series.
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