HW 3-solutions

HW 3-solutions - morris(gmm643 – HW 3 – mann –(54675 1 This print-out should have 20 questions Multiple-choice questions may continue on the

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Unformatted text preview: morris (gmm643) – HW 3 – mann – (54675) 1 This print-out should have 20 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. Sections 12.4, 12.5, 12.6 001 10.0 points If a m , b m , and c m satisfy the inequalities < b m ≤ c m ≤ a m , for all m , what can we say about the series ( A ) : ∞ summationdisplay m =1 a m , ( B ) : ∞ summationdisplay m =1 b m if we know that the series ( C ) : ∞ summationdisplay m = 1 c m is convergent but know nothing else about a m and b m ? 1. ( A ) diverges , ( B ) diverges 2. ( A ) diverges , ( B ) converges 3. ( A ) converges , ( B ) diverges 4. ( A ) need not converge , ( B ) converges correct 5. ( A ) converges , ( B ) converges 6. ( A ) converges , ( B ) need not converge Explanation: Let’s try applying the Comparison Test: (i) if < b m ≤ c m , summationdisplay m c m converges , then the Comparison Test applies and says that summationdisplay b m converges; (ii) while if < c m ≤ a m , summationdisplay m c m converges , then the Comparison Test is inconclusive be- cause summationdisplay a m could converge, but it could di- verge - we can’t say precisely without further restrictions on a m . Consequently, what we can say is ( A ) need not converge , ( B ) converges . 002 10.0 points Determine whether the series ∞ summationdisplay k = 1 1 3 radicalbig k ( k + 1)( k + 4) converges or diverges. 1. series is divergent correct 2. series is convergent Explanation: Note first that lim k →∞ k 3 k ( k + 1)( k + 4) = 1 > . Thus by the limit comparison test, the given series summationdisplay k = ∞ 1 3 radicalbig k ( k + 1)( k + 4) converges if and only if the series ∞ summationdisplay k =1 1 k converges. But by the p-series test with p = 1 (or use the comparison test applied to the har- monic series), this last series diverges. Conse- quently, the given series is divergent . 003 10.0 points morris (gmm643) – HW 3 – mann – (54675) 2 Determine whether the series ∞ summationdisplay k =1 k ( k + 3)3 k converges or diverges. 1. series is convergent correct 2. series is divergent Explanation: We use the Limit Comparison Test with a k = k ( k + 3)3 k , b k = 1 3 k . For lim k →∞ a k b k = lim k →∞ k k + 3 = 1 > . Thus the series ∞ summationdisplay k =1 k ( k + 3)3 k converges if and only if the series ∞ summationdisplay k =1 1 3 k converges. But this last series is a geometric series with | r | = 1 3 < 1 , hence convergent. Consequently, the given series is series is convergent . 004 10.0 points Which of the following series ( A ) ∞ summationdisplay n =1 6 n 3 n 2 + 5 ( B ) ∞ summationdisplay n =1 parenleftbigg 3 4 parenrightbigg n ( C ) ∞ summationdisplay n =24 parenleftbigg 3 4 parenrightbigg n converge(s)?...
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This note was uploaded on 11/17/2010 for the course PHY 56630 taught by Professor Coker during the Spring '10 term at University of Texas at Austin.

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HW 3-solutions - morris(gmm643 – HW 3 – mann –(54675 1 This print-out should have 20 questions Multiple-choice questions may continue on the

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