HW 13 key

# HW 13 key - Miller Kierste – Homework 13 – Due 3:00 am...

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Unformatted text preview: Miller, Kierste – Homework 13 – Due: Apr 24 2007, 3:00 am – Inst: Gary Berg 1 This print-out should have 20 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 Which, if any, of the following statements are true? A. If X a n converges, then lim n →∞ a n = 0. B. The Ratio Test can be used to determine whether X 1 /n 3 converges. C. If 0 ≤ a n ≤ b n and X b n diverges, then X a n diverges 1. A and C only 2. A and B only 3. none of them 4. all of them 5. B only 6. C only 7. B and C only 8. A only correct Explanation: A. True. To say that X a n converges is to say that the limit lim n →∞ s n of its partial sums s n = a 1 + a 2 + ... + a n converges. But then lim n →∞ a n = s n- s n- 1 = 0 . B. False: when a n = 1 /n 3 , then fl fl fl fl a n +1 a n fl fl fl fl = n 3 ( n + 1) 3-→ 1 as n → , ∞ , so the Ratio Test is inconclu- sive. C. False: set a n = 1 n 2 , b n = 1 n . Then 0 ≤ a n ≤ b n , but the Integral Test shows that X a n converges while X b n diverges. keywords: 002 (part 1 of 1) 10 points Which one of the following properties does the series ∞ X n = 3 (- 1) n- 1 n- 2 n 2 + n- 4 have? 1. divergent 2. absolutely convergent 3. conditionally convergent correct Explanation: The given series has the form ∞ X n = 3 (- 1) n- 1 n- 1 n 2 + n- 4 = ∞ X n = 3 (- 1) n- 1 f ( n ) where f is defined by f ( x ) = x- 2 x 2 + x- 4 . Notice that x 2 + x- 4 > 0 on [3 , ∞ ), so the terms in the given series are defined for all Miller, Kierste – Homework 13 – Due: Apr 24 2007, 3:00 am – Inst: Gary Berg 2 n ≥ 3. On the other hand, x- 2 > 0 on (2 , ∞ ), so x > 2 = ⇒ f ( x ) > . Now, by the Quotient Rule, f ( x ) = ( x 2 + x- 4)- ( x- 2)(2 x + 1) ( x 2 + x- 4) 2 =- x 2- 4 x + 2 ( x 2 + x- 4) 2 ; in particular, f is decreasing on [6 , ∞ ). Thus by the Limit Comparison Test and the p-series Test with p = 1, we see that the series ∞ X n = 6 f ( n ) diverges, so the given series fails to be abso- lutely convergent. But n ≥ 6 = ⇒ f ( n ) > f ( n + 1) , while lim x →∞ f ( x ) = 0 . Consequently, by The Alternating Series Test, the given series is conditionally convergent . keywords: 003 (part 1 of 1) 10 points Determine which, if any, of the series A. ∞ X m = 3 m + 2 ( m ln m ) 2 B. 1 + 1 2 + 1 4 + 1 8 + 1 16 + ... are divergent. 1. neither of them correct 2. B only 3. both of them 4. A only Explanation: A. Convergent: use Limit Comparison Test and Integral Test with f ( x ) = 1 x (ln x ) 2 . B. Convergent: given series is a geometric series ∞ X n = 0 ar n with a = 1 and r = 1 2 < 1. keywords: 004 (part 1 of 1) 10 points Which of the following properties does the series ∞ X k = 1 (- 6) k +1 4 3 k 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.

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HW 13 key - Miller Kierste – Homework 13 – Due 3:00 am...

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