This preview shows pages 1–4. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 = lim k  a k  1 /k has to be determined. Compute the value of for the series summationdisplay k = 1 3 k k (ln k + 8) k . 1. = 2. = 3 3. = 24 4. = 8 5. = 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, = 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...
View Full
Document
 Fall '08
 Cepparo

Click to edit the document details