Homework 14 - Rehman (aar638) – HW14 – sachse –...

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Unformatted text preview: Rehman (aar638) – HW14 – sachse – (56620) 1 This print-out should have 22 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Which, if any, of the following statements are true? A. If summationdisplay a n is divergent, then summationdisplay | a n | is divergent. B. The Ratio Test can be used to determine whether summationdisplay 1 /n ! converges. C. If lim n →∞ a n = 0, then summationdisplay a n converges. 1. none of them 2. all of them 3. A and C only 4. B and C only 5. A only 6. A and B only correct 7. C only 8. B only Explanation: A. True: if summationdisplay | a n | were convergent, then summationdisplay a n would be absolutely convergent, hence convergent. B. True: when a n = 1 /n !, then vextendsingle vextendsingle vextendsingle vextendsingle a n +1 a n vextendsingle vextendsingle vextendsingle vextendsingle = 1 n + 1-→ as n → , ∞ , so summationdisplay a n is convergent by Ratio Test. C. False: when a n = 1 /n , then lim n →∞ a n = 0, but ∞ summationdisplay n = 1 a n = ∞ summationdisplay n =1 1 n diverges by the Integral Test. 002 10.0 points Which one of the following properties does the series ∞ summationdisplay k =3 (- 1) k − 1 k- 1 k 2 + k- 2 have? 1. absolutely convergent 2. divergent 3. conditionally convergent correct Explanation: The given series has the form ∞ summationdisplay k = 3 (- 1) k − 1 k- 1 k 2 + k- 2 = ∞ summationdisplay k = 3 (- 1) k − 1 f ( k ) where f is defined by f ( x ) = x- 1 x 2 + x- 2 . Notice that x 2 + x- 2 > 0 on [3 , ∞ ), so the terms in the given series are defined for all k ≥ 3. On the other hand, x- 1 > 0 on (1 , ∞ ), so x > 1 = ⇒ f ( x ) > . Now, by the Quotient Rule, f ′ ( x ) = ( x 2 + x- 2)- ( x- 1)(2 x + 1) ( x 2 + x- 2) 2 =- x 2- 2 x + 1 ( x 2 + x- 2) 2 ; Rehman (aar638) – HW14 – sachse – (56620) 2 in particular, f is decreasing on [3 , ∞ ). Thus by the Limit Comparison Test and the p-series Test with p = 1, we see that the series ∞ summationdisplay k =3 f ( k ) diverges, so the given series fails to be abso- lutely convergent. But k ≥ 3 = ⇒ f ( k ) > f ( k + 1) , while lim x →∞ f ( x ) = 0 . Consequently, by The Alternating Series Test, the given series is conditionally convergent . 003 10.0 points Determine which, if any, of the series A. ∞ summationdisplay m = 3 m + 2 ( m ln m ) 2 B. 1 + 1 2 + 1 4 + 1 8 + 1 16 + . . . are convergent. 1. A only 2. neither of them 3. B only 4. both of them correct 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 se- ries ∞ summationdisplay n =0 ar n with a = 1 and r = 1 2 < 1....
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This note was uploaded on 02/01/2011 for the course MATH 408L taught by Professor Gogolev during the Fall '09 term at University of Texas.

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Homework 14 - Rehman (aar638) – HW14 – sachse –...

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