Chapter 7: Q.1.a) (page 630)

If $0 \leq f (x) \leq g (x)$ for every x $\geq$ 0 and the improper integral $\int_{0}^{\infty} g (x) d x$ converges, then the improper integral $\int_{0}^{\infty} f (x) d x$ converges. The objective is to whether determine the statement is true or false

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Step by step solution

To determine given statement is true or false

The improper integral $\int_{0}^{\infty} g (x) d x$ is convergent .Therefore ,

$\int_{0}^{\infty} g (x) d x = A$ ,where A is finite.

Given that $0 \leq f (x) \leq g (x)$ .

Therefore, $0 \leq \int_{0}^{\infty} f (x) d x \leq \int_{0}^{\infty} g (x) d x$

$\int_{0}^{\infty} f (x) d x \leq A$ .

Therefore, the improper integral $\int_{0}^{\infty} f (x) d x$ converges.

Therefore ,the given statement is true.

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If $0 \leq f (x) \leq g (x)$ for every x $\geq$ 0 and the improper integral $\int_{0}^{\infty} g (x) d x$ converges, then the improper integral $\int_{0}^{\infty} f (x) d x$ converges. The objective is to whether determine the statement is true or false

Short Answer

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To determine given statement is true or false

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If 0≤f(x)≤g(x)for every x≥0 and the improper integral∫0∞g(x)dxconverges, then the improper integral ∫0∞f(x)dxconverges. The objective is to whether determine the statement is true or false

Short Answer

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To determine given statement is true or false

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Most popular questions from this chapter

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If $0 \leq f (x) \leq g (x)$ for every x $\geq$ 0 and the improper integral $\int_{0}^{\infty} g (x) d x$ converges, then the improper integral $\int_{0}^{\infty} f (x) d x$ converges. The objective is to whether determine the statement is true or false