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HW 04-solutions - gonzales(pag757 – HW 04 – Odell...

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Unformatted text preview: gonzales (pag757) – HW 04 – Odell – (57000) 1 This print-out should have 26 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 properties does the series ∞ summationdisplay n =1 ( − 1) n- 1 5 n 2 + 3 3 n have? 1. conditionally convergent 2. absolutely convergent correct 3. divergent Explanation: The given series has the form ∞ summationdisplay n = 1 ( − 1) n- 1 b n , b n = 5 n 2 + 3 3 n of an alternating series. But the denominator is increasing very fast, so first let’s check if the series is absolutely convergent rather than simply conditionally convergent. We use the Ratio test, for then vextendsingle vextendsingle vextendsingle ( − 1) n b n +1 ( − 1) n- 1 b n vextendsingle vextendsingle vextendsingle = b n +1 b n = 1 3 5( n + 1) 2 + 3 5 n 2 + 3 . But 5( n + 1) 2 + 3 5 n 2 + 3 = 5 n 2 + 10 n + 8 5 n 2 + 3 −→ 1 as n → ∞ . Thus lim n →∞ vextendsingle vextendsingle vextendsingle ( − 1) n b n +1 ( − 1) n- 1 b n vextendsingle vextendsingle vextendsingle = 1 3 < 1 . Consequently, by the Ratio test, the given series is absolutely convergent . 002 10.0 points Determine whether the series ∞ summationdisplay n = 0 ( − 2) 2 n (2 n )! is absolutely convergent, conditionally con- vergent, or divergent. 1. divergent 2. conditionally convergent 3. absolutely convergent correct Explanation: We use the Ratio Test with a n = ( − 2) 2 n (2 n )! . For then vextendsingle vextendsingle vextendsingle a n +1 a n vextendsingle vextendsingle vextendsingle = 2 2( n +1) (2 n + 2)! (2 n )! 2 2 n = 2 2 (2 n + 1)(2 n + 2) . Thus lim n →∞ vextendsingle vextendsingle vextendsingle a n +1 a n vextendsingle vextendsingle vextendsingle = lim n →∞ 2 2 (2 n + 1)(2 n + 2) = 0 < 1 . Consequently, the series is absolutely convergent . 003 10.0 points Which one of the following properties does the series ∞ summationdisplay n =1 ( − 1) n 2 n + 3 have? gonzales (pag757) – HW 04 – Odell – (57000) 2 1. absolutely convergent 2. conditionally convergent correct 3. divergent Explanation: 004 10.0 points Which one of the following properties does the series ∞ summationdisplay n = 1 ( − 1) n 8 1 /n 2 n 3 + 5 have? 1. divergent 2. conditionally convergent 3. absolutely convergent correct Explanation: The given series is an alternating series ∞ summationdisplay n = 1 ( − 1) n a n where a n = 8 1 /n 2 n 3 + 5 ≤ 8 n 3 . But by the p-series test, the series ∞ summationdisplay n =1 8 n 3 is convergent. Consequently, by the Compar- ison test we see that the given series is absolutely convergent . 005 10.0 points Which one of the following properties does the series ∞ summationdisplay n = 1 ( − 1) n +1 (5 n 2 + 1)6 n n !...
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