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Unformatted text preview: pokharel (yp624) HW12 Radin (57410) 1 This printout should have 21 questions. Multiplechoice 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 + 1 n + 2 , what is a n when n 2? 1. a n = 7 n ( n + 2) 2. a n = 7 ( n + 2)( n + 3) 3. a n = 7 ( n + 2)( n + 1) 4. a n = 5 ( n + 2)( n + 3) 5. a n = 5 ( n + 2)( n + 1) correct 6. a n = 5 n ( n + 2) 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 + 1 n + 2 3( n 1) + 1 ( n 1) + 2 . Consequently, a n = 5 ( n + 2)( n + 1) . 002 10.0 points Determine whether the series summationdisplay n = 0 4 (cos n ) parenleftbigg 1 2 parenrightbigg n is convergent or divergent, and if convergent, find its sum. 1. convergent with sum 8 3 2. convergent with sum 8 3. divergent 4. convergent with sum 3 8 5. convergent with sum 8 6. convergent with sum 8 3 correct Explanation: Since cos n = ( 1) n , the given series can be rewritten as an infinite geometric series summationdisplay n =0 4 parenleftbigg 1 2 parenrightbigg n = summationdisplay n = 0 a r n in which a = 4 , r = 1 2 . 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 3 . 003 10.0 points pokharel (yp624) HW12 Radin (57410) 2 Determine whether the series summationdisplay n = 1 3 n 2 4 n 2 + 1 is convergent or divergent, and if convergent, find its sum. 1. convergent with sum = 3 5 2. convergent with sum = 3 4 3. divergent correct 4. convergent with sum = 5 3 5. convergent with sum = 4 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 = 3 n 2 4 n 2 + 1 = 3 4 + 1 n 2 , so lim n a n = lim n 3 n 2 4 n 2 + 1 = 3 4 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 = 1 4 3. converges with sum = 1 4. converges with sum = 5 6 correct 5. converges with sum = 1 2 6. series diverges 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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 Fall '08
 RAdin

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