HW12-solutions - cadena(jc59484 HW12 lawn(55930 This print-out should have 20 questions Multiple-choice questions may continue on the next column or

HW12-solutions - cadena(jc59484 HW12 lawn(55930 This...

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Unformatted text preview: cadena (jc59484) – HW12 – 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 one of the following series is conver- gent? 1. ∞ summationdisplay n = 1 ( − 1) 3 6 3 + √ n 2. ∞ summationdisplay n = 1 ( − 1) 2 n 6 3 + √ n 3. ∞ summationdisplay n = 1 ( − 1) n − 1 2 + √ n correct 4. ∞ summationdisplay n = 1 3 6 + √ n 5. ∞ summationdisplay n = 1 ( − 1) n − 1 2 + √ n 6 + √ n Explanation: Since ∞ summationdisplay n =1 ( − 1) 3 6 3 + √ n = − ∞ summationdisplay n =1 6 3 + √ n , use of the Limit Comparison and p-series Tests with p = 1 2 shows that this series is divergent. Similarly, since ∞ summationdisplay n =1 ( − 1) 2 n 6 3 + √ n = ∞ summationdisplay n =1 6 3 + √ n , the same argument shows that this series as well as ∞ summationdisplay n =1 3 6 + √ n is divergent. On the other hand, by the Divergence Test, the series ∞ summationdisplay n = 1 ( − 1) n − 1 2 + √ n 6 + √ n is divergent because lim n →∞ ( − 1) n − 1 2 + √ n 6 + √ n negationslash = 0 . This leaves only the series ∞ summationdisplay n = 1 ( − 1) n − 1 2 + √ n . To see that this series is convergent, set b n = 1 2 + √ n . Then (i) b n +1 ≤ b n , (ii) lim n →∞ b n = 0 . Consequently, by the Alternating Series Test, the series ∞ summationdisplay n = 1 ( − 1) n − 1 2 + √ n is convergent. 002 (part 1 of 3) 10.0 points Decide whether the series ∞ summationdisplay n = 1 ( − 1) n − 1 3 n − 1 6 n − 4 converges or diverges. 1. converges 2. diverges correct Explanation: The given series has the form ∞ summationdisplay n = 1 ( − 1) n − 1 f ( n ) with f defined by f ( x ) = 3 x − 1 6 x − 4 . cadena (jc59484) – HW12 – lawn – (55930) 2 Now lim x →∞ f ( x ) = 1 2 . Consequently, by the Divergence Test, the given series diverges . 003 (part 2 of 3) 10.0 points Decide whether the series ∞ summationdisplay n = 1 ( − 1) n − 1 3 n 2 6 n 3 + 4 converges or diverges. 1. converges correct 2. diverges Explanation: The given series has the form ∞ summationdisplay n = 1 ( − 1) n f ( n ) with f defined by f ( x ) = 3 x 2 6 x 3 + 4 . Now, by the Quotient Rule, f ′ ( x ) = 6(6 x 3 + 4) − 54 x 4 (6 x 3 + 4) 2 = − 3 x braceleftBig 6 x 3 − 4 (6 x 3 + 4) 2 bracerightBig . Thus f ′ ( x ) < 0 for all large x , so f ( n ) > f ( n + 1) for all large n . On the other hand, lim x →∞ 3 x 2 6 x 3 + 4 = 0 . Consequently, by the Alternating Series Test, the given series converges . 004 (part 3 of 3) 10.0 points Decide whether the series ∞ summationdisplay n = 1 ( − 1) n − 1 cos parenleftbigg 1 n parenrightbigg converges or diverges....
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  • Fall '11
  • Gramlich
  • Accounting, Mathematical Series, lim, n=1

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