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Unformatted text preview: Create assignment, 59725, Review 3, Dec 02 at 2:19 pm 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. CalC12a21s 55:01, calculus3, multiple choice, < 1 min, wordingvariable. 001 Determine whether the sequence { a n } con verges or diverges when a n = ( 1) n 1 n n 2 + 7 , and if it converges, find the limit. 1. converges with limit = 0 correct 2. converges with limit = 1 7 3. converges with limit = 7 4. converges with limit = 1 7 5. sequence diverges 6. converges with limit = 7 Explanation: After division, a n = ( 1) n 1 n n 2 + 7 = ( 1) n 1 n + 1 n . Consequently, ≤  a n  = 1 n + 1 n ≤ 1 n . But 1 /n → 0 as n → ∞ , so by the Squeeze theorem, lim n →∞  a n  = 0 . But a n  ≤ a n ≤  a n  , so by the Squeeze theorem again the given sequence { a n } converges and has limit = 0 . keywords: CalC12a35c 55:01, calculus3, multiple choice, > 1 min, wordingvariable. 002 Determine if the sequence { a n } converges, and if it does, find its limit when a n = µ 1 5 3 n ¶ 4 n . 1. limit = e 20 3 correct 2. limit = e 5 3 3. limit = e 15 4 4. limit = e 5 3 5. the sequence diverges 6. limit = e 20 3 7. limit = 1 Explanation: We know that lim n →∞ ‡ 1 x n · n = e x . Now a n = •µ 1 5 3 n ¶ n ‚ 4 . By the properties of limits, therefore, lim n →∞ a n = • lim n →∞ µ 1 5 3 n ¶ n ‚ 4 , Create assignment, 59725, Review 3, Dec 02 at 2:19 pm 2 so the sequence converges and has limit = ‡ e 5 3 · 4 = e 20 3 . keywords: CalC12b12exam 55:02, calculus3, multiple choice, > 1 min, wordingvariable. 003 Determine whether the series 3 9 2 + 27 4 81 8 + ··· is convergent or divergent, and if convergent, find its sum. 1. series is divergent correct 2. convergent with sum = 3 5 3. convergent with sum = 2 4. convergent with sum = 3 5. convergent with sum = 4 5 Explanation: The infinite series 3 9 2 + 27 4 81 8 + ··· = ∞ X n = 1 a r n 1 is an infinite geometric series with a = 3 , r = 3 2 . But 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 . Consequently, the given series is divergent . keywords: infinite series, geometric series, di vergent CalC12b15exam 55:02, calculus3, multiple choice, > 1 min, wordingvariable. 004 Determine whether the series ∞ X n = 0 3 µ 4 5 ¶ n is convergent or divergent, and if convergent, find its sum. 1. convergent, sum = 15 correct 2. divergent 3. convergent, sum = 16 4. convergent, sum = 5 3 5. convergent, sum = 16 Explanation: The given series is an infinite geometric series ∞ X n = 0 a r n with a = 3 and r = 4 5 . But the sum of such a series is (i) convergent with sum a 1 r when  r  < 1, (ii) divergent when  r  ≥ 1. Create assignment, 59725, Review 3, Dec 02 at 2:19 pm 3 Consequently, the given series is convergent, sum = 15 .....
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 Fall '07
 CHENG
 Mathematical Series, lim

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