HW13-solutions - cadena(jc59484 HW13 lawn(55930 This print-out should have 20 questions Multiple-choice questions may continue on the next column or

HW13-solutions - cadena(jc59484 HW13 lawn(55930 This...

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Unformatted text preview: cadena (jc59484) – HW13 – lawn – (55930) 1 This print-out should have 20 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 0 ≤ a n ≤ b n and summationdisplay b n diverges, then summationdisplay a n diverges B. The Ratio Test can be used to determine whether summationdisplay 1 /n ! converges. C. If summationdisplay a n converges, then lim n →∞ a n = 0. 1. A only 2. B and C only correct 3. A and B only 4. none of them 5. all of them 6. A and C only 7. B only 8. C only Explanation: A. 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. 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. 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 m =2 (- 1) m m m 2 + 2 have? 1. absolutely convergent 2. divergent 3. conditionally convergent correct Explanation: The given series has the form ∞ summationdisplay m =2 (- 1) m m m 2 + 2 = ∞ summationdisplay m =2 (- 1) m f ( m ) where f ( x ) = x x 2 + 2 . To check whether f ( x ) is decreasing, we look at f ′ ( x ). By the Quotient Rule, f ′ ( x ) = ( x 2 + 2)- 2 x 2 ( x 2 + 2) 2 =- x 2- 2 ( x 2 + 2) 2 < on [2 , ∞ ). Thus f ( x ) is decreasing on [2 , ∞ ). On the other hand, lim x →∞ f ( x ) = lim x →∞ x x 2 + 2 = 0 . cadena (jc59484) – HW13 – lawn – (55930) 2 So by The Alternating Series Test, the series ∞ summationdisplay m =2 (- 1) m m m 2 + 2 converges. Does it converge absolutely? Well, ∞ summationdisplay m =2 vextendsingle vextendsingle vextendsingle vextendsingle (- 1) m m m 2 + 2 vextendsingle vextendsingle vextendsingle vextendsingle = ∞ summationdisplay m =2 m m 2 + 2 . But this last series diverges by the p-series and Limit Comparison Tests. Consequently, the series ∞ summationdisplay m =2 (- 1) m m m 2 + 2 is conditionally convergent . 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. neither A nor B 2. A only 3. both A and B 4. B only correct Explanation: (A) FALSE: set a n = (- 1) n n ....
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  • Fall '11
  • Gramlich
  • Accounting, Mathematical Series, lim

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