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Unformatted text preview: Martinez, Gorge – Exam 3 – Due: Dec 4 2007, 11:00 pm – Inst: Diane Radin 1 This printout should have 18 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 Determine whether the sequence { a n } con verges or diverges when a n = ( 1) n µ 3 n + 4 3 n + 3 ¶ , and if it does, find its limit. 1. limit = 4 3 2. limit = ± 1 3. sequence diverges correct 4. limit = 0 5. limit = 1 Explanation: After division, 3 n + 4 3 n + 3 = 3 + 4 n 3 + 3 n . Now 4 n , 3 n → 0 as n → ∞ , so lim n →∞ 3 n + 4 3 n + 3 = 1 6 = 0 . Thus as n → ∞ , the values of a n oscillate be tween values ever closer to ± 1. Consequently, the sequence diverges . keywords: 002 (part 1 of 1) 10 points Determine if the sequence { a n } converges when a n = n 6 n ( n 1) 6 n , and if it does, find its limit 1. limit = e 6 2. limit = e 1 6 3. limit = e 6 correct 4. sequence diverges 5. limit = e 1 6 6. limit = 1 Explanation: By the Laws of Exponents, a n = µ n 1 n ¶ 6 n = µ 1 1 n ¶ 6 n = h‡ 1 1 n · n i 6 . But ‡ 1 + x n · n→ e x as n → ∞ . Consequently, { a n } converges and has limit = ( e 1 ) 6 = e 6 . keywords: sequence, e, exponentials, limit 003 (part 1 of 1) 10 points Determine whether the series 1 + 3 + 9 + 27 + ··· is convergent or divergent, and if convergent, find its sum. 1. convergent with sum = 2 9 Martinez, Gorge – Exam 3 – Due: Dec 4 2007, 11:00 pm – Inst: Diane Radin 2 2. convergent with sum = 9 2 3. convergent with sum = 1 2 4. convergent with sum = 2 5. divergent correct Explanation: The series 1 + 3 + 9 + 27 + ··· = ∞ X n =1 a r n 1 is an infinite geometric series in which a = 1 and r = 3. But such a series is (i) convergent with sum a 1 r when  r  < 1, (ii) divergent when  r  ≥ 1 . Thus the given series is divergent . keywords: 004 (part 1 of 1) 10 points Determine whether the series ∞ X n = 0 2 (cos nπ ) µ 3 4 ¶ n is convergent or divergent, and if convergent, find its sum. 1. convergent with sum 8 7 correct 2. convergent with sum 8 7 3. divergent 4. convergent with sum 8 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 ∞ X n =0 2 µ 3 4 ¶ n = ∞ X 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 . keywords: geometric series, convergent 005 (part 1 of 1) 10 points Determine whether the infinite series ∞ X n =1 3( n + 1) 2 n ( n + 2) converges or diverges, and if converges, find its sum....
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 Fall '09
 RAdin
 Calculus, Mathematical Series, Martinez, Diane Radin

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