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Unformatted text preview: tgo72 Homework 11 Gompf (58370) 1 This printout should have 19 questions. Multiplechoice questions may continue on the next column or page find all choices before answering. 001 10.0 points Find the sum of the finite series 3 + 3 7 9 + 3 7 2 9 2 + . . . + 3 7 6 9 6 . 1. sum = 3 9 6 parenleftBig 9 7 7 7 2 parenrightBig correct 2. sum = 3 parenleftBig 9 6 7 6 2 parenrightBig 3. sum = 3 9 6 parenleftBig 9 6 7 6 2 parenrightBig 4. sum = 3 9 6 parenleftBig 9 5 7 5 2 parenrightBig 5. sum = 3 parenleftBig 9 7 7 7 2 parenrightBig Explanation: The given series is a finite geometric series 6 summationdisplay n = 0 ar n , with a = 3 , r = 7 9 . Now 6 summationdisplay n = 0 ar n = a parenleftBig 1 r 7 1 r parenrightBig . Consequently, sum = 3 9 6 parenleftBig 9 7 7 7 2 parenrightBig . 002 10.0 points Rewrite the series 2 parenleftbigg 3 8 parenrightbigg 2 sin 3 5 2 parenleftbigg 3 8 parenrightbigg 3 sin 4 6 + 2 parenleftbigg 3 8 parenrightbigg 4 sin 5 7 + . . . using summation notation. 1. sum = summationdisplay k = 1 parenleftbigg 3 8 parenrightbigg k 2 sin( k + 2) k + 4 2. sum = summationdisplay k = 3 parenleftbigg 3 8 parenrightbigg k 1 2 sin k k + 1 3. sum = summationdisplay k = 3 parenleftbigg 3 8 parenrightbigg k 1 2 sin k k + 2 correct 4. sum = 60 summationdisplay k = 2 parenleftbigg 3 8 parenrightbigg k 2 sin( k + 1) k + 3 5. sum = 20 summationdisplay k = 3 parenleftbigg 3 8 parenrightbigg k 1 2 sin k k + 2 Explanation: The given series is an infinite series, so two of the answers must be incorrect because they are finite series written in summation notation. Starting summation at k = 3 we see that the general term of the infinite series is a k = 2 parenleftbigg 3 8 parenrightbigg k 1 sin k k + 2 . Consequently, sum = summationdisplay k = 3 parenleftbigg 3 8 parenrightbigg k 1 2 sin k k + 2 . 003 10.0 points If the n th partial sum of an infinite series is S n = 4 n 2 5 5 n 2 + 2 , what is the sum of the series? 1. sum = 3 4 2. sum = 13 20 tgo72 Homework 11 Gompf (58370) 2 3. sum = 7 10 4. sum = 4 5 correct 5. sum = 3 5 Explanation: By definition sum = lim n S n = lim n parenleftBig 4 n 2 5 5 n 2 + 2 parenrightBig . Thus sum = 4 5 . 004 10.0 points Determine whether the series summationdisplay n = 0 2 (cos n ) parenleftbigg 1 2 parenrightbigg n is convergent or divergent, and if convergent, find its sum. 1. convergent with sum 4 2. convergent with sum 3 4 3. convergent with sum 4 3 4. convergent with sum 4 3 correct 5. divergent 6. convergent with sum 4 Explanation: Since cos n = ( 1) n , the given series can be rewritten as an infinite geometric series summationdisplay n =0 2 parenleftbigg 1 2 parenrightbigg n = summationdisplay n = 0 a r n in which a = 2 , r = 1 2 ....
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 Spring '08
 RAdin
 Calculus

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