Q. 12 What, if anything, does the dive... [FREE SOLUTION]

Chapter 5: Q. 12 (page 478)

What, if anything, does the divergence of ${鈭�}_{1}^{鈭�} \frac{1}{x^{2}} d x$ and the comparison test tell you about the convergence or divergence of ${鈭�}_{1}^{鈭�} \frac{1}{x^{2} + 1} d x$ and why?

Short Answer

Expert verified

The convergence of ${鈭�}_{1}^{鈭�} \frac{1}{x^{2}} d x$ and comparison test are useful to tell the convergence of ${鈭�}_{1}^{鈭�} \frac{1}{x^{2} + 1} d x$ .

Step by step solution

Step 1. Given information.

We are given two integrals,

{鈭�}_{1}^{鈭�} \frac{1}{x^{2}} d x

and

${鈭�}_{1}^{鈭�} \frac{1}{x^{2} + 1} d x$

Step 2. Comparing the integrals.

Finding the relation between $\frac{1}{x^{2}}$ and $\frac{1}{x^{2} + 1}$ in the interval $[1, 鈭�)$ .

$x^{2} < x^{2} + 1$ for all $x 鈭� [1, 鈭�)$

$\frac{1}{x^{2}} > \frac{1}{x^{2} + 1}$ for all $x 鈭� [1, 鈭�)$

That is, $\frac{1}{x^{2} + 1} < \frac{1}{x^{2}}$ for all $x 鈭� [1, 鈭�)$

Apply comparison test for improper integrals to the continuous functions, we get

Since ${鈭�}_{1}^{鈭�} \frac{1}{x^{2}} d x$ converges and $\frac{1}{x^{2} + 1} < \frac{1}{x^{2}}$ in the interval $[1, 鈭�)$ , the improper integral ${鈭�}_{1}^{鈭�} \frac{1}{x^{2} + 1} d x$ must converge in the interval $[1, 鈭�)$ .

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91影视

What, if anything, does the divergence of ${鈭�}_{1}^{鈭�} \frac{1}{x^{2}} d x$ and the comparison test tell you about the convergence or divergence of ${鈭�}_{1}^{鈭�} \frac{1}{x^{2} + 1} d x$ and why?

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Comparing the integrals.

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91影视

What, if anything, does the divergence of 鈭�1鈭�1x2dxand the comparison test tell you about the convergence or divergence of 鈭�1鈭�1x2+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

Logic and Functions

Mechanics Maths

Calculus

Probability and Statistics

Decision Maths

Statistics

Study anywhere. Anytime. Across all devices.

What, if anything, does the divergence of ${鈭�}_{1}^{鈭�} \frac{1}{x^{2}} d x$ and the comparison test tell you about the convergence or divergence of ${鈭�}_{1}^{鈭�} \frac{1}{x^{2} + 1} d x$ and why?