Exam3Sol

# Exam3Sol - Massaro Michael – Exam 3 – Due Dec 5 2007...

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Unformatted text preview: Massaro, Michael – Exam 3 – Due: Dec 5 2007, 1:00 am – Inst: Shinko Harper 1 This print-out should have 18 questions. Multiple-choice 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 7 n + 3 ¶ , and if it does, find its limit. 1. limit = 3 7 2. limit = ± 3 7 3. limit = 0 4. sequence diverges correct 5. limit = 4 3 Explanation: After division, 3 n + 4 7 n + 3 = 3 + 4 n 7 + 3 n . Now 4 n , 3 n → 0 as n → ∞ , so lim n →∞ 3 n + 4 7 n + 3 = 3 7 6 = 0 . Thus as n → ∞ , the values of a n oscillate be- tween values ever closer to ± 3 7 . Consequently, the sequence diverges . keywords: 002 (part 1 of 1) 10 points Determine if the sequence { a n } converges when a n = n 4 n ( n- 8) 4 n , and if it does, find its limit 1. limit = e- 32 2. limit = e 32 correct 3. limit = e- 2 4. sequence diverges 5. limit = e 2 6. limit = 1 Explanation: By the Laws of Exponents, a n = µ n- 8 n ¶- 4 n = µ 1- 8 n ¶- 4 n = h‡ 1- 8 n · n i- 4 . But ‡ 1 + x n · n-→ e x as n → ∞ . Consequently, { a n } converges and has limit = ( e- 8 )- 4 = e 32 . keywords: sequence, e, exponentials, limit 003 (part 1 of 1) 10 points Determine whether the series 2 + 3 + 9 2 + 27 4 + ··· is convergent or divergent, and if convergent, find its sum. 1. convergent with sum = 1 9 Massaro, Michael – Exam 3 – Due: Dec 5 2007, 1:00 am – Inst: Shinko Harper 2 2. convergent with sum = 4 3. convergent with sum = 1 4 4. divergent correct 5. convergent with sum = 9 Explanation: The series 2 + 3 + 9 2 + 27 4 + ··· = ∞ X n =1 a r n- 1 is an infinite geometric series in which a = 2 and r = 3 2 . 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 1 2 ¶ n is convergent or divergent, and if convergent, find its sum. 1. convergent with sum 4 2. divergent 3. convergent with sum- 4 4. convergent with sum- 3 4 5. convergent with sum- 4 3 6. convergent with sum 4 3 correct Explanation: Since cos nπ = (- 1) n , the given series can be rewritten as an infinite geometric series ∞ X n =0 2 µ- 1 2 ¶ n = ∞ X n = 0 a r n in which a = 2 , 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 4 3 . 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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Exam3Sol - Massaro Michael – Exam 3 – Due Dec 5 2007...

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