   Chapter 7.R, Problem 11TFQ

Chapter
Section
Textbook Problem

# Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If f is a continuous, decreasing function on [ 1 , ∞ ) and lim x → ∞ f ( x ) = 0 , then ∫ 1 ∞ f ( x )   d x is convergent.

To determine

To Check: True/False

If f is continuous, decreasing function on [1,) and limxf(x)=0, then 1f(x)dx is convergent.

Explanation

The statement is false. To justify, consider the counter example that is f(x)=1x.

Here, the function is decreasing on [1,) and limxf(x)=limx1x=0. Now,

1f(x)dx=1

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