Homework 11

# Homework 11 - tgo72 – Homework 11 – Gompf –(58370 1...

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Unformatted text preview: tgo72 – Homework 11 – Gompf – (58370) 1 This print-out should have 19 questions. Multiple-choice 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 nπ ) 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 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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Homework 11 - tgo72 – Homework 11 – Gompf –(58370 1...

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