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Unformatted text preview: moseley (cmm3869) HW04 Gilbert (56380) 1 This printout should have 24 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 =0 4 (3) n is convergent or divergent, and if convergent, find its sum. 1. convergent, sum = 5 2 2. convergent, sum = 5 2 3. convergent, sum = 1 4. divergent correct 5. convergent, sum = 2 Explanation: The given series is an infinite geometric series summationdisplay n = 0 a r n with a = 4 and r = 3. But the sum of such a series is (i) convergent with sum a 1 r when  r  < 1, (ii) divergent when  r  1. Consequently, the given series is divergent . 002 10.0 points Find the sum of the infinite series summationdisplay k = 1 (cos 2 ) k , (0 < 2 ) , whenever the series converges. 1. sum = cot 2 correct 2. sum = csc 2 3. sum = tan 2 4. sum = sec 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 . 003 10.0 points Determine whether the series summationdisplay n = 1 n 3 5 n 3 + 3 is convergent or divergent, and if convergent, find its sum. 1. convergent with sum = 8 2. convergent with sum = 1 5 3. convergent with sum = 1 8 4. convergent with sum = 5 5. divergent correct moseley (cmm3869) HW04 Gilbert (56380) 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 = n 3 5 n 3 + 3 = 1 5 + 3 n 3 , so lim n a n = lim n n 3 5 n 3 + 3 = 1 5 negationslash = 0 . Thus the given series is divergent . 004 10.0 points Determine if the series summationdisplay n = 1 2 + 3 n 4 n converges or diverges, and if it converges, find its sum. 1. converges with sum = 3 2. converges with sum = 8 3 3. series diverges 4. converges with sum = 7 3 5. converges with sum = 11 3 correct 6. converges with sum = 10 3 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 2 4 n = summationdisplay n = 1 1 2 parenleftBig 1 4 parenrightBig n 1 is a geometric series with a = r = 1 4 < 1. Thus it converges with sum = 2 3 , while summationdisplay n = 1 3 n 4 n = summationdisplay n = 1 3 4 parenleftBig 3 4 parenrightBig n 1 is a geometric series with a = r = 3 4 < 1....
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This note was uploaded on 04/12/2011 for the course M 408d taught by Professor Sadler during the Spring '07 term at University of Texas at Austin.
 Spring '07
 Sadler
 Multivariable Calculus

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