Answered: If 0 < G(t) < H (t) for 0

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter0: A Review Of Basic Algebra

Section0.2: Integer Exponents And Scientific Notation

Problem 89E

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decide if the given statement is true or false,and give a brief justiﬁcation for your answer.If true, you can quote a relevant deﬁnition or theorem . If false,provide an example,illustration,or briefexplanation of why the statement is false.

If 0 < G(t) < H (t) for 0 <t < o and
| G()dt
converges, then so does
H(t)dt.

Expert Solution

Step 1

Given information:

$0 \leq G (t) \leq H (t)$ for $0 \leq t \leq \infty$ and $\int_{0}^{\infty} G (t) d t$ converges.

Explanation:

Suppose $F (t) = \int_{0}^{\infty} G (t) d t$ and $Q (t) = \int_{0}^{\infty} H (t) d t$ .

The improper integral $F (t) = \int_{0}^{\infty} G (t) d t$ and $Q (t) = \int_{0}^{\infty} H (t) d t$ can be written in the form of series as $F (t) = \sum_{t = 0}^{\infty} G (t), Q (t) = \sum_{t = 0}^{\infty} H (t)$ .

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