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Unformatted text preview: Portillo, Tom – Exam 3 – Due: Dec 5 2006, 11:00 pm – Inst: David Benzvi 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 series ∞ X n = 2 ( 1) n n 2 ln n is conditionally convergent, absolutely con vergent, or divergent. 1. series is absolutely convergent 2. series is divergent correct 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 2 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: 002 (part 1 of 1) 10 points Which one of the following properties does the series ∞ X n = 3 ( 1) n 3 n (ln n ) n have? 1. divergent 2. absolutely convergent correct 3. conditionally convergent Explanation: The given series can be written in the form ∞ X n = 3 ( 1) n 3 n (ln n ) n = ∞ X n = 3 ( 1) n a n with a n = 3 n (ln n ) n > . Now < a n +1 a n = 3(ln n ) n (ln( n + 1)) n +1 = 3 ‡ ln n ln( n + 1) · n n 1 ln( n + 1) o < 3 ln( n + 1) . Consequently, lim n →∞ a n +1 a n = 0 . In view of the Ratio Test, therefore, the series ∞ X n = 3 fl fl fl ( 1) n 3 n (ln n ) n fl fl fl converges, so the given series is absolutely convergent . Portillo, Tom – Exam 3 – Due: Dec 5 2006, 11:00 pm – Inst: David Benzvi 2 keywords: 003 (part 1 of 1) 10 points Which one of the following properties does the series ∞ X n = 1 ( 3) n n ! have? 1. absolutely convergent correct 2. divergent 3. conditionally convergent Explanation: The given series has the form ∞ X n = 1 a n , a n = ( 3) n n ! . But then fl fl fl a n +1 a n fl fl fl = 3( n !) ( n + 1)! = 3 n + 1 , in which case lim n →∞ fl fl fl a n +1 a n fl fl fl = 0 < 1 . Consequently, by the Ratio test, the given series is absolutely convergent . keywords: alternating series, absolutely con vergent, divergent, conditionally convergent, Ratio Test 004 (part 1 of 1) 10 points Which one of the following properties does the series ∞ X m = 3 ( 1) m 1 m 2 m 2 + m 3 have? 1. conditionally convergent correct 2. divergent 3. absolutely convergent Explanation: The given series has the form ∞ X m = 3 ( 1) m 1 m 1 m 2 + m 3 = ∞ X m = 3 ( 1) m 1 f ( m ) where f is defined by f ( x ) = x 2 x 2 + x 3 . Notice that x 2 + x 3 > 0 on [3 , ∞ ), so the terms in the given series are defined for all m ≥ 3. On the other hand, x 2 > 0 on (2 , ∞ ), so x > 2 = ⇒ f ( x ) > ....
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 Spring '06
 Benzvi
 Mathematical Series, Portillo, David Benzvi

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