This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: moore (jwm2685) HW03 gilbert (55485) 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 Determine whether the series summationdisplay n = 1 2 n 3 4 n 3 + 1 is convergent or divergent, and if convergent, find its sum. 1. convergent with sum = 2 5 2. convergent with sum = 2 3. divergent correct 4. convergent with sum = 1 2 5. convergent with sum = 5 2 Explanation: The infinite series summationdisplay n =1 a n is divergent when lim n a n exists but lim n a n negationslash = 0 . Note for the given series, a n = 2 n 3 4 n 3 + 1 = 2 4 + 1 n 3 , so lim n a n = lim n 2 n 3 4 n 3 + 1 = 1 2 negationslash = 0 . Thus the given series is divergent . 002 10.0 points Determine if the series summationdisplay n = 1 4 + 2 n 5 n converges or diverges, and if it converges, find its sum. 1. converges with sum = 19 12 2. converges with sum = 7 4 3. converges with sum = 17 12 4. converges with sum = 5 3 correct 5. series diverges 6. converges with sum = 3 2 Explanation: An infinite geometric series n =1 a r n 1 (i) converges when  r  < 1 and has sum = a 1 r , while it (ii) diverges when  r  1 . Now summationdisplay n = 1 4 5 n = summationdisplay n = 1 4 5 parenleftBig 1 5 parenrightBig n 1 is a geometric series with a = r = 1 5 < 1. Thus it converges with sum = 1 , while summationdisplay n = 1 2 n 5 n = summationdisplay n = 1 2 5 parenleftBig 2 5 parenrightBig n 1 is a geometric series with a = r = 2 5 < 1. Thus it too converges, and it has sum = 2 3 . moore (jwm2685) HW03 gilbert (55485) 2 Consequently, being the sum of two conver gent series, the given series converges with sum = 1 + 2 3 = 5 3 . 003 10.0 points Find the sum of the infinite series summationdisplay k = 1 (cos 2 ) k , (0 < 2 ) , whenever the series converges. 1. sum = sec 2 2. sum = tan 2 3. sum = cot 2 correct 4. sum = csc 2 5. sum = sin 2 cos 2 Explanation: For general the series summationdisplay k = 1 (cos 2 ) k is an infinite geometric series with common ratio cos 2 . Since the series starts at k = 1, its sum is thus given by cos 2 1 cos 2 = cos 2 sin 2 . Consequently sum = cot 2 . 004 10.0 points Let g be a continuous, positive, decreasing function on [1 , ). Compare the values of the series A = 13 summationdisplay n = 1 g ( n ) and the integral B = integraldisplay 14 1 g ( z ) dz . 1. A = B 2. A > B correct 3. A < B Explanation: In the figure 1 2 3 4 5 . . . a 1 a 2 a 3 a 4 the bold line is the graph of g on [1 , ) and the areas of the rectangles the terms in the series summationdisplay n = 1 a n , a n = g ( n ) ....
View Full
Document
 Spring '07
 TextbookAnswers

Click to edit the document details