dx S₁ (In(x))²2* The work you do in this question is to establish the convergence or divergence of this improper integral using the comparison test. 1. Consider the integral I = (a) Show that 1 ≤ ln(x) ≤ x for all x ≥ 4. Do this by considering their relative sizes at x = 4 (the "starting line") and then considering which grows faster for x>4. The latter must be done by considering the derivatives of ln(x) and x for x>4. Comparing the initial (that is, at x = 4) values of ln(x) and x and comparing their derivatives for x>4 will inform you about which function is greater than the other for x> 4. This should establish the chain of inequalities. You need to provide the explanation. You may not simply refer to a graph of the two functions. (b) As a result of the previous part, it follows that 1 ≤ (ln(x))² ≤ x². (You don't need to provide and argument to justify this but you should convince yourself it is true). Explain why these inequal- ities are not helpful for determining why I converges. (Hint: consider the relationship between the reciprocals of these functions.) (c) Show that 1 ≤ln(x) ≤ √x for all x ≥4. Use the same approach as in part (a). (d) Use the inequalities in (c) to establish that the given integral diverges.
dx S₁ (In(x))²2* The work you do in this question is to establish the convergence or divergence of this improper integral using the comparison test. 1. Consider the integral I = (a) Show that 1 ≤ ln(x) ≤ x for all x ≥ 4. Do this by considering their relative sizes at x = 4 (the "starting line") and then considering which grows faster for x>4. The latter must be done by considering the derivatives of ln(x) and x for x>4. Comparing the initial (that is, at x = 4) values of ln(x) and x and comparing their derivatives for x>4 will inform you about which function is greater than the other for x> 4. This should establish the chain of inequalities. You need to provide the explanation. You may not simply refer to a graph of the two functions. (b) As a result of the previous part, it follows that 1 ≤ (ln(x))² ≤ x². (You don't need to provide and argument to justify this but you should convince yourself it is true). Explain why these inequal- ities are not helpful for determining why I converges. (Hint: consider the relationship between the reciprocals of these functions.) (c) Show that 1 ≤ln(x) ≤ √x for all x ≥4. Use the same approach as in part (a). (d) Use the inequalities in (c) to establish that the given integral diverges.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please explain, thank you
Transcribed Image Text:1. Consider the integral I =
The work you do in this question is to establish the convergence
or divergence of this improper integral using the comparison test.
dx
(ln(x))²
(a) Show that 1 ≤ ln(x) ≤ x for all x ≥ 4. Do this by considering their relative sizes at x = 4 (the
"starting line") and then considering which grows faster for x> 4. The latter must be done by
considering the derivatives of In(x) and x for x>4. Comparing the initial (that is, at x = 4) values
of ln(x) and x and comparing their derivatives for x > 4 will inform you about which function is
greater than the other for x > 4. This should establish the chain of inequalities. You need to
provide the explanation. You may not simply refer to a graph of the two functions.
(b) As a result of the previous part, it follows that 1 ≤ (ln(x))² ≤ x². (You don't need to provide and
argument to justify this but you should convince yourself it is true). Explain why these inequal-
ities are not helpful for determining why I converges. (Hint: consider the relationship between
the reciprocals of these functions.)
(c) Show that 1 ≤ln(x) ≤ √x for all x ≥ 4. Use the same approach as in part (a).
(d) Use the inequalities in (c) to establish that the given integral diverges.
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