HW11-solutions - cadena(jc59484 HW11 lawn(55930 This print-out should have 17 questions Multiple-choice questions may continue on the next column or

HW11-solutions - cadena(jc59484 HW11 lawn(55930 This...

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cadena (jc59484) – HW11 – lawn – (55930) 1 This print-out should have 17 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Let g be a continuous, positive, decreasing function on [4 , ). Compare the values of the integral A = 19 4 g ( t ) dt and the series B = 19 n = 5 g ( n ) . 1. A > B correct 2. A < B 3. A = B Explanation: In the figure 4 5 6 7 8 . . . a 5 a 6 a 7 a 8 the bold line is the graph of g on [4 , ) and the areas of the rectangles the terms in the series n = 5 a n , a n = g ( n ) . Clearly from this figure we see that a 5 = g (5) < 5 4 g ( t ) dt, a 6 = g (6) < 6 5 g ( t ) dt , while a 7 = g (7) < 7 6 g ( t ) dt, a 8 = g (8) < 8 7 g ( t ) dt , and so on. Consequently, A > B . keywords: Szyszko 002 10.0 points Which, if any, of the following series con- verge? A . 1 + 1 4 + 1 9 + 1 16 + . . . B . n = 1 n 1 + n 2 1. both of them 2. A only correct 3. B only 4. neither of them Explanation: A. Series is n =1 1 n 2 . Use f ( x ) = 1 x 2 . Then 1 f ( x ) dx is convergent, so series converges.
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cadena (jc59484) – HW11 – lawn – (55930) 2 B. Use f ( x ) = x 1 + x 2 . Then 1 f ( x ) dx is divergent (ln integral), so series diverges. 003 10.0 points Determine whether the series n = 1 6 - 2 n n 3 converges or diverges. 1. series is divergent 2. series is convergent correct Explanation: We check separately the convergence or di- vergence of n = 1 6 n 3 , n = 1 2 n n 3 = n = 1 2 n 5 / 2 . Now both are p-series: the first being of the form n = 1 6 n 3 = n = 1 6 n p , p = 3 , while the second one has the form n = 1 2 n 5 / 2 = n = 1 2 n p , p = 5 2 . Since p > 1 in both cases, each series con- verges. As the di ff erence of convergent series, therefore, the given series is convergent . 004 10.0 points Determine whether the series n = 2 n 8(ln( n )) 2 is convergent or divergent. 1. convergent 2. divergent correct Explanation: By the Divergence Test, a series n = N a n will be divergent for each fixed choice of N if lim n → ∞ a n = 0 since it is only the behaviour of a n as n → ∞ that’s important. Now, for the given series, N = 2 and a n = n 8(ln( n )) 2 . But by L’Hospital’s Rule applied twice, lim x → ∞ x (ln( x )) 2 = lim x → ∞ 1 (2 ln( x )) /x = lim x → ∞ x 2 ln( x ) = lim x → ∞ 1 2 /x = . Consequently, by the Divergence Test, the given series is divergent . 005 10.0 points If the improper integral 1 1 x p dx converges, which of the following statements is (are) always true? ( A ) n 1 n p converges; ( B ) n 1 n p +1 diverges;
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cadena (jc59484) – HW11 – lawn – (55930) 3 ( C ) n 1 n p - 1 converges; ( D ) n 1 n p - 1 diverges; ( E ) n 1 n p +1 converges . 1. A and E only correct 2. B and D only 3. A, C and E only 4. A, D and E 5. A only Explanation: To apply the Integral test we need to start with a function f which is positive, continuous and decreasing on [1 , ). Then the integral test says that the improper integral 1 f ( x ) dx converges if and only if the infinite series n = 1 f ( n ) converges.
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