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Unformatted text preview: fh2825 – Homework 11 – Radin – (58415) 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 6 + 6 · 5 8 + 6 · 5 2 8 2 + . . . + 6 · 5 7 8 7 . 1. sum = 6 8 7 parenleftBig 8 6 − 5 6 3 parenrightBig 2. sum = 6 parenleftBig 8 8 − 5 8 3 parenrightBig 3. sum = 6 8 7 parenleftBig 8 8 − 5 8 3 parenrightBig correct 4. sum = 6 8 7 parenleftBig 8 7 − 5 7 3 parenrightBig 5. sum = 6 parenleftBig 8 7 − 5 7 3 parenrightBig Explanation: The given series is a finite geometric series 7 summationdisplay n = 0 ar n , with a = 6 , r = 5 8 . Now 7 summationdisplay n = 0 ar n = a parenleftBig 1 − r 8 1 − r parenrightBig . Consequently, sum = 6 8 7 parenleftBig 8 8 − 5 8 3 parenrightBig . 002 10.0 points Rewrite the series 8 parenleftbigg 4 5 parenrightbigg 2 sin 3 5 − 8 parenleftbigg 4 5 parenrightbigg 3 sin 4 6 + 8 parenleftbigg 4 5 parenrightbigg 4 sin 5 7 + . . . using summation notation. 1. sum = 15 summationdisplay k = 3 parenleftbigg − 4 5 parenrightbigg k − 1 8 sin k k + 2 2. sum = 70 summationdisplay k = 2 parenleftbigg 4 5 parenrightbigg k 8 sin( k + 1) k + 3 3. sum = ∞ summationdisplay k = 3 parenleftbigg − 4 5 parenrightbigg k − 1 8 sin k k + 1 4. sum = ∞ summationdisplay k = 1 parenleftbigg − 4 5 parenrightbigg k 8 sin( k + 2) k + 4 5. sum = ∞ summationdisplay k = 3 parenleftbigg − 4 5 parenrightbigg k − 1 8 sin k k + 2 correct 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 = 8 parenleftbigg − 4 5 parenrightbigg k − 1 sin k k + 2 . Consequently, sum = ∞ summationdisplay k = 3 parenleftbigg − 4 5 parenrightbigg k − 1 8 sin k k + 2 . 003 10.0 points If the n th partial sum of an infinite series is S n = 3 n 2 − 1 n 2 + 2 , what is the sum of the series? 1. sum = 11 4 2. sum = 5 2 fh2825 – Homework 11 – Radin – (58415) 2 3. sum = 2 4. sum = 3 correct 5. sum = 9 4 Explanation: By definition sum = lim n →∞ S n = lim n →∞ parenleftBig 3 n 2 − 1 n 2 + 2 parenrightBig . Thus sum = 3 . 004 10.0 points Determine whether the series ∞ summationdisplay n = 0 2 (cos nπ ) parenleftbigg 1 2 parenrightbigg n is convergent or divergent, and if convergent, find its sum. 1. divergent 2. convergent with sum − 4 3. convergent with sum − 4 3 4. convergent with sum 4 3 correct 5. convergent with sum 4 6. convergent with sum − 3 4 Explanation: Since cos nπ = ( − 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
 Geometric Series, lim

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