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HW14-solutions

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adams (bda255) – HW14 – Radin – (56635) 1 This print-out should have 22 questions. Multiple-choice questions may continue on the next column or page – fnd all choices beFore answering. 001 10.0 points Which, iF any, oF the Following statements are true? A. IF s a n is divergent, then s | a n | is divergent. B. The Ratio Test can be used to determine whether s 1 /n ! converges. C. IF lim n →∞ a n = 0, then s a n converges. 1. A and C only 2. none oF them 3. A only 4. C only 5. B only 6. A and B only correct 7. all oF them 8. B and C only Explanation: A. True: iF s | a n | were convergent, then s a n would be absolutely convergent, hence convergent. B. True: when a n = 1 /n !, then v v v v a n +1 a n v v v v = 1 n + 1 -→ 0 as n , , so s a n is convergent by Ratio Test. C. ±alse: when a n = 1 /n , then lim n →∞ a n = 0, but s n = 1 a n = s n = 1 1 n diverges by the Integral Test. 002 10.0 points Which one oF the Following properties does the series s k = 2 ( - 1) k 1 k - 3 k 2 + k - 4 have? 1. absolutely convergent 2. conditionally convergent correct 3. divergent Explanation: The given series has the Form s k = 2 ( - 1) k 1 k - 1 k 2 + k - 4 = s k = 2 ( - 1) k 1 f ( k ) where f is defned 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 defned For all k 2. On the other hand, x - 3 > 0 on (3 , ), so x > 3 = f ( x ) > 0 . 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 ;

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adams (bda255) – HW14 – Radin – (56635) 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 s 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. 1 + 1 2 + 1 4 + 1 8 + 1 16 + . . . B. s m = 3 m + 2 ( m ln m ) 2 are convergent. 1. both of them correct 2. B only 3. A only 4. neither of them Explanation: A. Convergent: given series is a geometric se- ries s n = 0 ar n with a = 1 and r = 1 2 < 1. B. Convergent: use Limit Comparison Test and Integral Test with f ( x ) = 1 x (ln x ) 2 . 004 10.0 points Determine whether the series s m = 1 ( - 3) m +1 2 3 m is absolutely convergent, conditionally con- vergent or divergent. 1. conditionally convergent 2. divergent 3. absolutely convergent correct Explanation: The given series can be written in the form s m = 1 ( - 3) m +1 2 3 m = s m = 1 a n with a m = ( - 1) m +1 3 m +1 2 3 m = ( - 1) m +1 3 p 3 2 3 P m .
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