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Unformatted text preview: Miller, Kierste – Review 3 – Due: May 10 2007, 7:00 pm – Inst: Gary Berg 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 Which, if any, of the following statements are true? A. The Ratio Test can be used to deter mine whether the series ∞ X n = 1 1 n 2 converges or diverges. B. The Root Test can be used to determine whether the series ∞ X k = 1 ‡ ln k 2 + k · k converges or diverges. 1. neither of them 2. both of them 3. A only 4. B only correct Explanation: A. False: when a n = 1 /n 2 , then fl fl fl fl a n +1 a n fl fl fl fl = ‡ n n + 1 · 2→ 1 as n → ∞ , so the Ratio Test is inconclusive. B. True: when a k = ‡ ln k 2 + k · k , then  a k  1 /k = ln k 2 + k→ as k → ∞ , so ∑ a k is convergent by the Root Test. keywords: 002 (part 1 of 1) 10 points Determine whether the infinite series ∞ X n = 1 4 n 3 n 7 n converges or diverges, and if it converges, find its sum. 1. converges with sum = 7 12 correct 2. series diverges 3. converges with sum = 2 3 4. converges with sum = 5 12 5. converges with sum = 1 6 6. converges with sum = 1 2 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 ∞ X n = 1 4 n 7 n = ∞ X n = 1 4 7 µ 4 7 ¶ n 1 is a geometric series with a = r = 4 7 < 1. Thus it converges with sum = 4 3 , Miller, Kierste – Review 3 – Due: May 10 2007, 7:00 pm – Inst: Gary Berg 2 while ∞ X n = 1 3 n 7 n = ∞ X n = 1 3 7 µ 3 7 ¶ n 1 is a geometric series with a = r = 3 7 < 1. Thus it too converges, and it has sum = 3 4 . Consequently, being the difference of two con vergent series, the given series converges with sum = 4 3 3 4 = 7 12 . keywords: infinite series, geometric series, sum, 003 (part 1 of 1) 10 points Determine whether the series ∞ X n = 2 ( 1) n n 6 ln n is conditionally convergent, absolutely con vergent, or divergent. 1. series is divergent correct 2. series is absolutely convergent 3. series is conditionally convergent Explanation: By the Divergence Test, a series ∞ X n = N ( 1) n a n will be divergent for each fixed choice of N if lim n →∞ a n 6 = 0 since it is only the behaviour of a n as n → ∞ that’s important. Now, for the given series, N = 2 and a n = n 6 ln n . But by L’Hospital’s Rule, lim x →∞ x ln x = lim x →∞ 1 1 /x = ∞ . Consequently, by the Divergence Test, the given series is divergent . keywords: 004 (part 1 of 1) 10 points Which one of the following properties does the series ∞ X m = 1 ( 1) m 1 3 + m 3 √ 2 + m 8 have?...
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This note was uploaded on 05/08/2008 for the course M 408 L taught by Professor Cepparo during the Spring '08 term at University of Texas at Austin.
 Spring '08
 Cepparo

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