HW 13 key - Miller, Kierste Homework 13 Due: Apr 24 2007,...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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?...
View Full Document

Page1 / 13

HW 13 key - Miller, Kierste Homework 13 Due: Apr 24 2007,...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online