HW13-solutions - kim(tk5895 – HW13 – Henry –(54974 1...

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Unformatted text preview: kim (tk5895) – HW13 – Henry – (54974) 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) n − 1 3 + √ n correct 2. ∞ summationdisplay n = 1 ( − 1) n − 1 3 + √ n 6 + √ n 3. ∞ summationdisplay n = 1 2 6 + √ n 4. ∞ summationdisplay n = 1 ( − 1) 2 n 6 2 + √ n 5. ∞ summationdisplay n = 1 ( − 1) 3 6 2 + √ n Explanation: Since ∞ summationdisplay n =1 ( − 1) 3 6 2 + √ n = − ∞ summationdisplay n =1 6 2 + √ 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 2 + √ n = ∞ summationdisplay n =1 6 2 + √ n , the same argument shows that this series as well as ∞ summationdisplay n =1 2 6 + √ n is divergent. On the other hand, by the Divergence Test, the series ∞ summationdisplay n = 1 ( − 1) n − 1 3 + √ n 6 + √ n is divergent because lim n →∞ ( − 1) n − 1 3 + √ n 6 + √ n negationslash = 0 . This leaves only the series ∞ summationdisplay n = 1 ( − 1) n − 1 3 + √ n . To see that this series is convergent, set b n = 1 3 + √ 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 3 + √ n is convergent. 002 10.0 points Determine whether the series ∞ summationdisplay n =0 4 n n 2 + 2 cos 2 nπ converges or diverges. 1. series converges 2. series diverges correct Explanation: Since cos 2 nπ = 1, the given series can be rewritten as the series ∞ summationdisplay n =0 4 n n 2 + 2 = ∞ summationdisplay n = 0 f ( n ) of positive, non-alternating terms where f ( x ) = 4 x x 2 + 2 . kim (tk5895) – HW13 – Henry – (54974) 2 Now lim x →∞ xf ( x ) = lim x →∞ 4 x 2 x 2 + 2 = 4 . Thus by the Limit Comparison Test, the given series converges if and only if the series ∞ summationdisplay n = 1 1 n converges. But, by applying the p-series Test with p = 1, we see that this last series di- verges. Consequently, the given series diverges . 003 10.0 points Determine whether the series ∞ summationdisplay n =0 4 ln( n + 5) cos parenleftBig nπ 2 parenrightBig converges or diverges. (Similar such series are basic to the mathematical description of all musical sounds!) 1. series diverges 2. series converges correct Explanation: The basic idea is that cos π 2 x is a peri- odic function taking values between − 1 and 1. This will enable us to write the given series as an alternating series. Indeed, cos 0 = 1 , cos π 2 = 0 , cos π = − 1 , cos 3 π 2 = 0 , and so on for higher multiples of π 2 . Thus the given series can be written as 4 braceleftBig 1 ln 5 − 1 ln7 + 1 ln9 − 1 ln 11 + . . ....
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This note was uploaded on 04/09/2011 for the course M 408 L taught by Professor Cepparo during the Fall '08 term at University of Texas.

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HW13-solutions - kim(tk5895 – HW13 – Henry –(54974 1...

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