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HW12-solutions

# HW12-solutions - kim(tk5895 – HW12 – Henry –(54974 1...

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Unformatted text preview: kim (tk5895) – HW12 – Henry – (54974) 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 Rewrite the series 5 parenleftbigg 3 7 parenrightbigg 2 sin 3 5- 5 parenleftbigg 3 7 parenrightbigg 3 sin 4 6 + 5 parenleftbigg 3 7 parenrightbigg 4 sin 5 7 + . . . using summation notation. 1. sum = ∞ summationdisplay k = 3 parenleftbigg- 3 7 parenrightbigg k − 1 5 sin k k + 2 correct 2. sum = ∞ summationdisplay k = 1 parenleftbigg- 3 7 parenrightbigg k 5 sin( k + 2) k + 4 3. sum = 40 summationdisplay k = 3 parenleftbigg- 3 7 parenrightbigg k − 1 5 sin k k + 2 4. sum = 20 summationdisplay k = 2 parenleftbigg 3 7 parenrightbigg k 5 sin( k + 1) k + 3 5. sum = ∞ summationdisplay k = 3 parenleftbigg- 3 7 parenrightbigg k − 1 5 sin k k + 1 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 = 5 parenleftbigg- 3 7 parenrightbigg k − 1 sin k k + 2 . Consequently, sum = ∞ summationdisplay k = 3 parenleftbigg- 3 7 parenrightbigg k − 1 5 sin k k + 2 . 002 10.0 points Determine whether the series ∞ summationdisplay n = 0 2 (cos nπ ) parenleftbigg 2 3 parenrightbigg n is convergent or divergent, and if convergent, find its sum. 1. convergent with sum- 5 6 2. convergent with sum 6 3. divergent 4. convergent with sum 6 5 correct 5. convergent with sum- 6 6. convergent with sum- 6 5 Explanation: Since cos nπ = (- 1) n , the given series can be rewritten as an infinite geometric series ∞ summationdisplay n =0 2 parenleftbigg- 2 3 parenrightbigg n = ∞ summationdisplay n = 0 a r n in which a = 2 , r =- 2 3 . But the series ∑ ∞ n =0 ar n is (i) convergent with sum a 1- r when | r | < 1, and (ii) divergent when | r | ≥ 1. Consequently, the given series is convergent with sum 6 5 . kim (tk5895) – HW12 – Henry – (54974) 2 003 10.0 points Determine if the series ∞ summationdisplay m = 1 2 + 3 m 5 m converges or diverges, and if it converges, find its sum. 1. converges with sum = 2 correct 2. converges with sum = 15 8 3. series diverges 4. converges with sum = 7 4 5. converges with sum = 13 8 6. converges with sum = 17 8 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 m = 1 2 5 m = ∞ summationdisplay m = 1 2 5 parenleftBig 1 5 parenrightBig m − 1 is a geometric series with a = r = 1 5 < 1. Thus it converges with sum = 1 2 , while ∞ summationdisplay m = 1 3 m 5 m = ∞ summationdisplay m = 1 3 5 parenleftBig 3 5 parenrightBig m − 1 is a geometric series with a = r = 3 5 < 1....
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HW12-solutions - kim(tk5895 – HW12 – Henry –(54974 1...

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