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# UT M 408D - Assignment

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abdallah haa2348 Assignment 8 Due Sun 4 12 by 11 59pm reyes 52535 This print out should have 39 questions Multiple choice questions may continue on the next column or page find all choices before answering 001 Clearly from this figure we see that a2 f 2 10 0 points a3 f 3 Compare the values of the series A X n 2 1 Z Z 2 f x dx 1 3 f x dx 2 while 2 n1 4 a4 f 4 and the improper integral Z 2 B dx x1 4 1 a5 f 5 Z Z 4 f x dx 3 5 f x dx 4 and so on for all n Consequently 1 A B correct 2 A B A X n 2 2 n1 4 Z 1 2 dx B x1 4 3 A B 002 Explanation In the figure 10 0 points Let g be a continuous positive decreasing function on 4 Compare the values of the integral A Z 17 g t dt 4 a2 1 a3 a4 2 3 and the series a5 4 5 B 16 X n 4 the bold line is the graph of the function f x 2 x1 4 on 1 and the area of each of the rectangles is one of the values of an 2 n1 4 1 A B correct 2 A B 3 A B Explanation In the figure g n abdallah haa2348 Assignment 8 Due Sun 4 12 by 11 59pm reyes 52535 2 and the integral B Z 13 f t dt 4 1 A B correct 2 A B a4 4 a5 5 a6 a7 6 7 8 3 A B Explanation In the figure the bold line is the graph of g on 4 and the areas of the rectangles the terms in the series X an an g n n 4 Clearly from this figure we see that Z 5 a4 g 4 g t dt 4 a5 g 5 Z a4 6 a5 a6 a7 g t dt 5 4 5 6 7 8 while a6 g 6 Z a7 g 7 Z 7 g t dt 6 8 g t dt 7 and so on Consequently A B the bold line is the graph of f on 4 and the areas of the rectangles the terms in the series X an an f n n 4 Clearly from this figure we see that f 4 Z 5 Z 6 Z 7 Z 8 f t dt 4 keywords 003 f 5 10 0 points Let f be a continuous positive decreasing function on 4 Compare the values of the series 12 X A f n n 4 f t dt 5 while f 6 f t dt 6 f 7 7 f t dt abdallah haa2348 Assignment 8 Due Sun 4 12 by 11 59pm reyes 52535 and so on Consequently 3 the solid blue line is the graph of g on 5 and the areas of the rectangles the terms in the series Clearly from this figure we see that A B g 5 keywords Z 6 Z 7 Z 8 Z 9 g x dx 5 004 10 0 points g 6 g x dx 6 Let g be a continuous positive decreasing function on 5 Compare the values of the integral Z 20 A g x dx while g 7 g 8 5 and the series B 20 X g n C n 6 19 X g n g x dx 7 g x dx 8 and so on On the other hand in the figure n 5 1 C A B 2 A B C 3 B C A a6 4 B C A 5 a7 6 a8 7 a9 8 9 5 A B C the solid blue line is again the graph of g on 5 but now 6 B A C correct Explanation In the figure a6 g 6 a7 g 7 Z Z 6 g x dx 5 7 g x dx 6 while a8 g 8 a5 5 a6 6 a7 7 a8 8 9 a9 g 9 Z Z 8 g x dx 7 9 g x dx 8 abdallah haa2348 Assignment 8 Due Sun 4 12 by 11 59pm reyes 52535 and so on Consequently B A C 005 10 0 points Which of the following series are convergent X A 3 n2 1 n 1 4 converges Thus for the three given series we have to use an appropriate choice of f 3 A Use f x 2 Then x 1 Z f x dx 1 is convergent tan 1 integral 2 B Use f x 2 3 Then x Z f x dx 1 X B n 1 C 1 2 n2 3 1 1 1 4 9 16 is divergent 1 Then x2 Z f x dx C Use f x 1 is convergent keywords convergent Integral test 1 all of them 006 2 B and C only 10 0 points Which if any of the following series converge 3 A and B only 4 none of them X A n 1 5 A only B 6 B only 1 n 1 n2 1 1 1 4 9 16 7 A and C only correct 8 C only Explanation By the Integral test if f x is a positive decreasing function then the infinite series X f n n 1 converges if and only if the improper integral Z f x dx 1 1 B only correct 2 A only 3 both of them 4 neither of them Explanation A Use f x x Then 1 x2 Z f x dx 1 abdallah haa2348 Assignment 8 Due Sun 4 12 by 11 59pm reyes 52535 is divergent ln integral so series diverges X 1 1 B Series is Use f x 2 Then 2 n x Note first that f is continuous and positive on 1 in addition since 2 f x e x 1 2x2 0 n 1 Z f x dx 1 for x 1 f is decreasing on 1 Thus we can use the Integral Test Now by substitution is convergent so series converges 007 Z 10 0 points Determine the convergence or divergence of the series A t xe B and so 2 1 e x 2 2 xe x dx t 1 1 2e 008 10 0 points 2 me m First find an so that X 1 A convergent B divergent n 1 2 A divergent B convergent 6 2 3 3 an 6 2 3 4 5 5 and then determine whether the series converges or diverges 3 both series convergent correct 1 an 3 series diverges 2n3 2 2 an 3 series converges 2n3 2 4 both series divergent Explanation A The given series has the form 1 1 1 1 4 9 16 25 X n 1 3 an 1 n2 This is a p series with p 2 1 so the series converges 4 an 5 an B The given series has the form X Z Since the integral converges the series converges m 1 1 dx 1 and X x2 1 …

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