Chapter 5: Q. 12 (page 478)

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

Short Answer

Expert verified

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

Step by step solution

Step 1. Given information.

We are given two integrals,

\int_{1}^{\infty} \frac{1}{x^{2}} d x

and

$\int_{1}^{\infty} \frac{1}{x^{2} + 1} d x$

Step 2. Comparing the integrals.

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

$x^{2} < x^{2} + 1$ for all $x \in [1, \infty)$

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

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

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

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

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

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Comparing the integrals.

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What, if anything, does the divergence of ∫1∞1x2dxand the comparison test tell you about the convergence or divergence of ∫1∞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

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