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hw14 - padilla(tp5647 – HW14 – Gilbert –(56650 1 This...

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Unformatted text preview: padilla (tp5647) – HW14 – Gilbert – (56650) 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. The Ratio Test can be used to determine whether summationdisplay 1 /n 3 converges. B. If 0 ≤ a n ≤ b n and summationdisplay b n diverges, then summationdisplay a n diverges C. If summationdisplay a n converges, then lim n →∞ a n = 0. 1. all of them 2. B and C only 3. A and B only 4. A only 5. A and C only 6. none of them 7. B only 8. C only correct Explanation: A. False: when a n = 1 /n 3 , then vextendsingle vextendsingle vextendsingle vextendsingle a n +1 a n vextendsingle vextendsingle vextendsingle vextendsingle = n 3 ( n + 1) 3-→ 1 as n → , ∞ , so the Ratio Test is inconclu- sive. B. False: set a n = 1 n 2 , b n = 1 n . Then 0 ≤ a n ≤ b n , but the Integral Test shows that summationdisplay a n converges while summationdisplay b n diverges. C. True. To say that summationdisplay 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 . 002 10.0 points Which one of the following properties does the series ∞ summationdisplay k =2 (- 1) k − 1 k- 3 k 2 + k- 4 have? 1. divergent 2. absolutely convergent 3. conditionally convergent correct Explanation: The given series has the form ∞ summationdisplay k = 2 (- 1) k − 1 k- 1 k 2 + k- 4 = ∞ summationdisplay k = 2 (- 1) k − 1 f ( k ) where f is defined by f ( x ) = x- 3 x 2 + x- 4 . Notice that x 2 + x- 4 > 0 on [2 , ∞ ), so the terms in the given series are defined for all k ≥ 2. On the other hand, x- 3 > 0 on (3 , ∞ ), so x > 3 = ⇒ f ( x ) > . padilla (tp5647) – HW14 – Gilbert – (56650) 2 Now, by the Quotient Rule, f ′ ( x ) = ( x 2 + x- 4)- ( x- 3)(2 x + 1) ( x 2 + x- 4) 2 =- x 2- 6 x + 1 ( x 2 + x- 4) 2 ; in particular, f is decreasing on [7 , ∞ ). Thus by the Limit Comparison Test and the p-series Test with p = 1, we see that the series ∞ summationdisplay k =7 f ( k ) diverges, so the given series fails to be abso- lutely convergent. But k ≥ 7 = ⇒ 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 + 3 m 2 ln m + 2 B. 1 + 1 2 + 1 4 + 1 8 + 1 16 + . . . are convergent. 1. both of them 2. B only correct 3. A only 4. neither of them Explanation: A. Divergent: use Limit Comparison Test and Integral Test with f ( x ) = 1 x ln x ....
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