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Unformatted text preview: kim (tk5895) HW13 Henry (54974) 1 This printout should have 20 questions. Multiplechoice 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 pseries 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 n converges or diverges. 1. series converges 2. series diverges correct Explanation: Since cos 2 n = 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, nonalternating 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 pseries 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 n 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 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 2 = 0 , cos = 1 , cos 3 2 = 0 , and so on for higher multiples of 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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 Fall '08
 Cepparo
 Calculus

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