CalcHW11Answers - nav277 Homework 11 Odell(58340 This...

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nav277 – Homework 11 – Odell – (58340) 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 · 5 8 + 3 · 5 2 8 2 + . . . + 3 · 5 7 8 7 . 1. sum = 3 8 7 parenleftBig 8 8 5 8 3 parenrightBig correct 2. sum = 3 parenleftBig 8 8 5 8 3 parenrightBig 3. sum = 3 parenleftBig 8 7 5 7 3 parenrightBig 4. sum = 3 8 7 parenleftBig 8 7 5 7 3 parenrightBig 5. sum = 3 8 7 parenleftBig 8 6 5 6 3 parenrightBig Explanation: The given series is a finite geometric series 7 summationdisplay n =0 ar n , with a = 3 , r = 5 8 . Now 7 summationdisplay n =0 ar n = a parenleftBig 1 r 8 1 r parenrightBig . Consequently, sum = 3 8 7 parenleftBig 8 8 5 8 3 parenrightBig . 002 10.0 points Rewrite the series 3 parenleftbigg 5 7 parenrightbigg 2 sin 3 5 3 parenleftbigg 5 7 parenrightbigg 3 sin 4 6 + 3 parenleftbigg 5 7 parenrightbigg 4 sin 5 7 + . . . using summation notation. 1. sum = summationdisplay k =1 parenleftbigg 5 7 parenrightbigg k 3 sin( k + 2) k + 4 2. sum = 50 summationdisplay k =2 parenleftbigg 5 7 parenrightbigg k 3 sin( k + 1) k + 3 3. sum = summationdisplay k =3 parenleftbigg 5 7 parenrightbigg k 1 3 sin k k + 1 4. sum = summationdisplay k =3 parenleftbigg 5 7 parenrightbigg k 1 3 sin k k + 2 correct 5. sum = 25 summationdisplay k =3 parenleftbigg 5 7 parenrightbigg k 1 3 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 = 3 parenleftbigg 5 7 parenrightbigg k 1 sin k k + 2 . Consequently, sum = summationdisplay k =3 parenleftbigg 5 7 parenrightbigg k 1 3 sin k k + 2 . 003 10.0 points If the n th partial sum of an infinite series is S n = 3 n 2 2 n 2 + 5 , what is the sum of the series? 1. sum = 9 4 2. sum = 13 4
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nav277 – Homework 11 – Odell – (58340) 2 3. sum = 3 correct 4. sum = 5 2 5. sum = 11 4 Explanation: By definition sum = lim n →∞ S n = lim n → ∞ parenleftBig 3 n 2 2 n 2 + 5 parenrightBig . Thus sum = 3 . 004 10.0 points Determine whether the series summationdisplay n =0 2 (cos ) parenleftbigg 3 4 parenrightbigg n is convergent or divergent, and if convergent, find its sum. 1. convergent with sum 8 7 correct 2. convergent with sum 8 3. convergent with sum 8 4. divergent 5. convergent with sum 8 7 6. convergent with sum 7 8 Explanation: Since cos = ( 1) n , the given series can be rewritten as an infinite geometric series summationdisplay n =0 2 parenleftbigg 3 4 parenrightbigg n = summationdisplay n =0 a r n in which a = 2 , r = 3 4 . But the series n =0 ar n is (i) convergent with sum a 1 r when | r | < 1, and (ii) divergent when | r | ≥ 1.
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