M408L hwk13 solutions

M408L hwk13 solutions - Pham, Quoc Homework 13 Due: Apr 24...

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Pham, Quoc – Homework 13 – Due: Apr 24 2007, 3:00 am – Inst: Eric Katerman 1 This print-out should have 20 questions. Multiple-choice questions may continue on the next column or page – fnd 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 0 a n b n and X b n diverges, then X a n diverges B. The Ratio Test can be used to determine whether X 1 /n 3 converges. C. IF X a n converges, then lim n →∞ a n = 0. 1. A and B only 2. A and C only 3. B and C only 4. A only 5. B only 6. C only correct 7. none oF them 8. all oF them Explanation: A. ±alse: 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. B. ±alse: when a n = 1 /n 3 , then f f f f a n +1 a n f f f f = n 3 ( n + 1) 3 -→ 1 as n , , so the Ratio Test is inconclu- sive. C. 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 . 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 - 3 have? 1. conditionally convergent correct 2. divergent 3. absolutely convergent Explanation: The given series has the Form X n = 3 ( - 1) n - 1 n - 1 n 2 + n - 3 = X n = 3 ( - 1) n - 1 f ( n ) where f is defned 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 defned For all
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Pham, Quoc – Homework 13 – Due: Apr 24 2007, 3:00 am – Inst: Eric Katerman 2 n 3. On the other hand, x - 2 > 0 on (2 , ), so x > 2 = f ( x ) > 0 . Now, by the Quotient Rule, f 0 ( x ) = ( x 2 + x - 3) - ( x - 2)(2 x + 1) ( x 2 + x - 3) 2 = - x 2 - 4 x + 1 ( x 2 + x - 3) 2 ; in particular, f is decreasing on [5 , ). Thus by the Limit Comparison Test and the p -series Test with p = 1, we see that the series X n = 5 f ( n ) diverges, so the given series fails to be abso- lutely convergent. But n 5 = 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. 1 + 1 2 + 1 4 + 1 8 + 1 16 + ... B. X m = 3 m + 3 m 2 ln m + 2 are divergent. 1. B only correct 2. both of them 3. A only 4. neither of them Explanation: A. Convergent: given series is a geometric series X n = 0 ar n with a = 1 and r = 1 2 < 1. B. Divergent: use Limit Comparison Test and Integral Test with f ( x ) = 1 x ln x . keywords: 004 (part 1 of 1) 10 points Which of the following properties does the series X m = 1 ( - 5) m +1 4 2 m have? 1.
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M408L hwk13 solutions - Pham, Quoc Homework 13 Due: Apr 24...

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