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Unformatted text preview: Kim, Jin – Homework 11 – Due: Nov 13 2007, 3:00 am – Inst: Diane Radin 1 This printout should have 22 questions. Multiplechoice questions may continue on the next column or page – find all choices before answering. The due time is Central time. 001 (part 1 of 1) 10 points If the n th partial sum of an infinite series is S n = 3 n 2 4 2 n 2 + 1 , what is the sum of the series? 1. sum = 9 8 2. sum = 3 2 correct 3. sum = 11 8 4. sum = 5 4 5. sum = 1 Explanation: By definition sum = lim n →∞ S n = lim n →∞ ‡ 3 n 2 4 2 n 2 + 1 · . Thus sum = 3 2 . keywords: partial sum, definition of series 002 (part 1 of 1) 10 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 = 17 n ( n + 4) 3. a n = 17 ( n + 4)( n + 3) 4. a n = 7 n ( n + 4) 5. a n = 17 ( n + 4)( n + 5) 6. a n = 7 ( n + 4)( n + 5) Explanation: By definition S n = n X 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) . keywords: partial sum, definition of series 003 (part 1 of 1) 10 points Determine whether the series 4 16 3 + 64 9 256 27 + ··· is convergent or divergent, and if convergent, find its sum. 1. series is divergent correct 2. convergent with sum = 9 7 3. convergent with sum = 4 4. convergent with sum = 3 Kim, Jin – Homework 11 – Due: Nov 13 2007, 3:00 am – Inst: Diane Radin 2 5. convergent with sum = 8 7 Explanation: The infinite series 4 16 3 + 64 9 256 27 + ··· = ∞ X n = 1 ar n 1 is an infinite geometric series with a = 4 , r = 4 3 . But an infinite geometric series ∑ ∞ n = 1 ar n 1 (i) converges when  r  < 1 and has sum = a 1 r while it (ii) diverges when  r  ≥ 1 . Consequently, the given series is divergent . keywords: infinite series, geometric series, di vergent 004 (part 1 of 1) 10 points Determine whether the series ∞ X n = 1 5 n 2 n 2 + 3 is convergent or divergent, and if convergent, find its sum. 1. convergent with sum = 1 5 2. convergent with sum = 5 3. divergent correct 4. convergent with sum = 4 5 5. convergent with sum = 5 4 Explanation: The infinite series ∞ X n =1 a n is divergent when lim n →∞ a n exists but lim n →∞ a n 6 = 0 . Note for the given series, a n = 5 n 2 n 2 + 3 = 5 1 + 3 n 2 , so lim n →∞ a n = lim n →∞ 5 n 2 n 2 + 3 = 5 6 = 0 . Thus the given series is divergent . keywords: 005 (part 1 of 1) 10 points Determine whether the series ∞ X n = 0 3 µ 1 5 ¶ n is convergent or divergent, and if convergent, find its sum. 1. divergent 2. convergent, sum = 5 2 3. convergent, sum = 15 4 correct 4. convergent, sum = 4 5. convergent, sum = 4 Kim, Jin – Homework 11 – Due: Nov 13 2007, 3:00 am – Inst: Diane Radin 3 Explanation: The given series is an infinite geometric series ∞ X n = 0 ar n with a = 3 and r = 1 5 . But the sum of such a series is (i) convergent with sum a 1 r when...
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 Spring '09
 Turner
 Mathematical Series, Diane Radin

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