Q. 11 What, if anything, does the dive... [FREE SOLUTION]

Chapter 5: Q. 11 (page 478)

What, if anything, does the divergence of ${鈭�}_{1}^{鈭�} \frac{1}{x} d x$ and the comparison test tell you about the convergence or divergence of ${鈭�}_{1}^{鈭�} \frac{1}{x + 1} d x$ and why?

Short Answer

Expert verified

The divergence of ${鈭�}_{1}^{鈭�} \frac{1}{x + 1} d x$ and comparison test are not sufficient to tell the convergence or divergence of ${鈭�}_{1}^{鈭�} \frac{1}{x} d x$ .

Step by step solution

Step 1. Given information.

We are given two integrals,

${鈭�}_{1}^{鈭�} \frac{1}{x} d x$ and

${鈭�}_{1}^{鈭�} \frac{1}{x + 1} d x$

Step 2. Comparing the integrals.

Finding the relation between x and $x + 1$ in the interval $[1, 鈭�)$ .

$x < x + 1$ for all $x 鈭� [1, 鈭�)$

$\frac{1}{x} < \frac{1}{x + 1}$ for all $x 鈭� [1, 鈭�)$

That is, $\frac{1}{x + 1} < \frac{1}{x}$ for all $x 鈭� [1, 鈭�)$

Since ${鈭�}_{1}^{鈭�} \frac{1}{x} d x$ diverges and $\frac{1}{x + 1} < \frac{1}{x}$ in the interval $[1, 鈭�)$ , the improper integral ${鈭�}_{1}^{鈭�} \frac{1}{x + 1} d x$ either converges or diverges.

Therefore, the divergence of the improper integral and comparison test are not sufficient to tell the convergence or divergence of the improper integral.

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What, if anything, does the divergence of ${鈭�}_{1}^{鈭�} \frac{1}{x} d x$ and the comparison test tell you about the convergence or divergence of ${鈭�}_{1}^{鈭�} \frac{1}{x + 1} d x$ and why?

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Comparing the integrals.

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What, if anything, does the divergence of 鈭�1鈭�1xdxand the comparison test tell you about the convergence or divergence of鈭�1鈭�1x+1dxand why?

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Comparing the integrals.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Probability and Statistics

Applied Mathematics

Decision Maths

Mechanics Maths

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

What, if anything, does the divergence of ${鈭�}_{1}^{鈭�} \frac{1}{x} d x$ and the comparison test tell you about the convergence or divergence of ${鈭�}_{1}^{鈭�} \frac{1}{x + 1} d x$ and why?