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calc hw 4 - lawrence(cdl678 – Homework 4 – ODELL...

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Unformatted text preview: lawrence (cdl678) – Homework 4 – ODELL – (56280) 1 This print-out should have 31 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 lim n →∞ a n = 0, then summationdisplay a n converges. B. If summationdisplay a n is divergent, then summationdisplay | a n | is divergent. C. The Ratio Test can be used to determine whether summationdisplay 1 /n 3 converges. 1. B and C only 2. A and B only 3. C only 4. all of them 5. A only 6. none of them 7. A and C only 8. B only correct Explanation: A. 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. B. True: if summationdisplay | a n | were convergent, then summationdisplay a n would be absolutely convergent, hence convergent. C. 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. 002 10.0 points Which, if any, of the following statements are true? A. The Ratio Test can be used to deter- mine whether the series summationdisplay n = ∞ 1 /n ! converges or diverges. B. The Root Test can be used to determine whether the series ∞ summationdisplay k = 1 k 3 + k 2 converges or diverges. 1. both of them 2. A only correct 3. B only 4. neither of them Explanation: A. 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 ∑ a n is convergent by Ratio Test. B. False: when a k = k 3 + k 2 , then ( | a k | ) 1 /k = k 1 /k / (1 + k 2 ) 1 /k −→ 1 lawrence (cdl678) – Homework 4 – ODELL – (56280) 2 so the Root Test is inconclusive. 003 10.0 points If ∑ n a n converges, which, if any, of the following statements are true: (A) summationdisplay n | a n | is convergent , (B) lim n →∞ a n = 0 . 1. B only correct 2. neither A nor B 3. A only 4. both A and B Explanation: (A) FALSE: set a n = ( − 1) n n . Then summationdisplay n | a n | = summationdisplay n 1 n , so by the p-series test with p = 1, the series summationdisplay | a n | diverges. On the other hand, summationdisplay n a n = summationdisplay n ( − 1) n n converges by the Alternating Series Test. (B) TRUE: to say that ∑ n 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 = lim n →∞ ( S n − S n − 1 ) = 0 ....
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