M408L_HW12 - hyun(hh7953 – HW12 – gogolev –(57440 1...

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Unformatted text preview: hyun (hh7953) – HW12 – gogolev – (57440) 1 This print-out should have 21 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points If the n th partial sum of ∑ ∞ n =1 a n is given by S n = 3 n + 5 n + 4 , what is a n when n ≥ 2? 1. a n = 7 ( n + 4)( n + 3) correct 2. a n = 7 n ( n + 4) 3. a n = 17 ( n + 4)( n + 5) 4. a n = 17 ( n + 4)( n + 3) 5. a n = 7 ( n + 4)( n + 5) 6. a n = 17 n ( n + 4) Explanation: By definition S n = n summationdisplay k → 1 a n = a 1 + a 2 + . . . + a n . Thus, for n ≥ 2, a n = S n − S n − 1 = 3 n + 5 n + 4 − 3( n − 1) + 5 ( n − 1) + 4 . Consequently, a n = 7 ( n + 4)( n + 3) . 002 10.0 points Determine whether the series ∞ summationdisplay n = 0 2 (cos nπ ) parenleftbigg 3 4 parenrightbigg n is convergent or divergent, and if convergent, find its sum. 1. convergent with sum 8 2. convergent with sum − 8 7 3. divergent 4. convergent with sum 8 7 correct 5. convergent with sum − 8 6. convergent with sum − 7 8 Explanation: Since cos nπ = ( − 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. Consequently, the given series is convergent with sum 8 7 . 003 10.0 points hyun (hh7953) – HW12 – gogolev – (57440) 2 Determine whether the series ∞ summationdisplay n = 1 n 2 3 n 2 + 5 is convergent or divergent, and if convergent, find its sum. 1. divergent correct 2. convergent with sum = 3 3. convergent with sum = 8 4. convergent with sum = 1 8 5. convergent with sum = 1 3 Explanation: The infinite series ∞ summationdisplay n =1 a n is divergent when lim n →∞ a n exists but lim n →∞ a n negationslash = 0 . Note for the given series, a n = n 2 3 n 2 + 5 = 1 3 + 5 n 2 , so lim n →∞ a n = lim n →∞ n 2 3 n 2 + 5 = 1 3 negationslash = 0 . Thus the given series is divergent . 004 10.0 points Determine whether the infinite series ∞ summationdisplay n = 1 3 n − 2 n 5 n converges or diverges, and if it converges, find its sum. 1. converges with sum = 2 3 2. converges with sum = 5 6 correct 3. series diverges 4. converges with sum = 1 2 5. converges with sum = 1 4 6. converges with sum = 1 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 n = 1 3 n 5 n = ∞ summationdisplay n = 1 3 5 parenleftbigg 3 5 parenrightbigg n − 1 is a geometric series with a = r = 3 5 < 1....
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This note was uploaded on 01/19/2010 for the course M 57440 taught by Professor Gogolev during the Fall '09 term at University of Texas.

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M408L_HW12 - hyun(hh7953 – HW12 – gogolev –(57440 1...

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