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Chapter 7: Q 1 TF (page 630)

what is the comparison test for improper integrals?

Short Answer

Expert verified

If $f (x) \geq g (x) \geq 0$ on the interval $[0, \infty)$ then

1.if $\int_{0}^{\infty} f (x) d x$ converges then so does $\int_{0}^{\infty} g (x) d x$ converges

2.if $\int_{0}^{\infty} g (x) d x$ diverges then so does $\int_{0}^{\infty} f (x) d x$ diverges

Step by step solution

01

The comparison test for improper integral

If $f (x) \geq g (x) \geq 0$ on the interval $[0, \infty)$ then

1.if $\int_{0}^{\infty} f (x) d x$ converges then so does $\int_{0}^{\infty} g (x) d x$ converges

2.if $\int_{0}^{\infty} g (x) d x$ diverges then so does $\int_{0}^{\infty} f (x) d x$ diverges

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Most popular questions from this chapter

In Exercises 48–51 find all values of p so that the series converges.

$\sum_{k = 1}^{\infty} \frac{\ln (k)}{k^{p}}$

Explain why the integral test may be used to analyze the given series and then use the test to determine whether the series converges or diverges.

$\sum_{k = 1}^{\infty} (3 k - 1)^{- 2 / 3}$

Find the values of x for which the series $\sum_{k = 0}^{\infty} {(\frac{\cos x}{2})}^{k}$ converges.

The contrapositive: What is the contrapositive of the implication “If A, then B.”?

Find the contrapositives of the following implications:

If a quadrilateral is a square, then it is a rectangle.

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) A divergent series $\sum_{k = 1}^{\infty} a_{k}$ in which $a_{k} \to 0$ .

(b) A divergent p-series.

(c) A convergent p-series.

See all solutions

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