HW 3-solutions

HW 3-solutions - husain (aih243) – HW 3 – mann –...

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Unformatted text preview: husain (aih243) – 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 divergent but know nothing else about a m and b m ? 1. ( A ) diverges , ( B ) converges 2. ( A ) converges , ( B ) need not converge 3. ( A ) diverges , ( B ) need not diverge correct 4. ( A ) converges , ( B ) diverges 5. ( A ) diverges , ( B ) diverges 6. ( A ) need not diverge , ( B ) diverges Explanation: Let’s try applying the Comparison Test: (i) if < b m ≤ c m , summationdisplay m c m diverges , then the Comparison Test is inconclusive be- cause summationdisplay b m could diverge, but it could con- verge - we can’t say precisely without further restrictions on b m ; (ii) while if < c m ≤ a m , summationdisplay m c m diverges , then the Comparison Test applies and says that summationdisplay a m diverges. Consequently, what we can say is ( A ) diverges , ( B ) need not diverge . 002 10.0 points Determine whether the series ∞ summationdisplay k = 1 6 3 radicalbig k ( k + 2)( k + 5) converges or diverges. 1. series is divergent correct 2. series is convergent Explanation: Note first that lim k →∞ k 3 k ( k + 2)( k + 5) = 1 > . Thus by the limit comparison test, the given series summationdisplay k = ∞ 6 3 radicalbig k ( k + 2)( k + 5) converges if and only if the series ∞ summationdisplay k =1 6 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 husain (aih243) – HW 3 – mann – (54675) 2 Determine whether the series ∞ summationdisplay k =1 k ( k + 5)4 k converges or diverges. 1. series is divergent 2. series is convergent correct Explanation: We use the Limit Comparison Test with a k = k ( k + 5)4 k , b k = 1 4 k . For lim k →∞ a k b k = lim k →∞ k k + 5 = 1 > . Thus the series ∞ summationdisplay k =1 k ( k + 5)4 k converges if and only if the series ∞ summationdisplay k =1 1 4 k converges. But this last series is a geometric series with | r | = 1 4 < 1 , hence convergent. Consequently, the given series is series is convergent . 004 10.0 points Which of the following series ( A ) ∞ summationdisplay n =1 3 n 6 n 2 + 2 ( B ) ∞ summationdisplay n =1 parenleftbigg 1 3 parenrightbigg n ( C ) ∞ summationdisplay n =21 parenleftbigg 6 7 parenrightbigg n converge(s)?...
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This note was uploaded on 11/30/2010 for the course M 408d taught by Professor Sadler during the Spring '07 term at University of Texas.

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HW 3-solutions - husain (aih243) – HW 3 – mann –...

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