   Chapter 7.R, Problem 13TFQ

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 ∫ a ∞ f ( x )   d x and ∫ a ∞ g ( x )   d x are both divergent, then ∫ a ∞ [ f ( x ) + g ( x ) ]   d x is divergent.

To determine

To Check: True/False

If af(x)dx and ag(x)dx are both divergent, then a[f(x)+g(x)]dx is divergent.

Explanation

The statement is false. To justify it, let’s take a counter example that is let f(x)=1x and g(x)=1x. Now, for the considered functions, the integrals af(x)dx and ag(x)dx are both divergent. Now,

a[f(x)+g(x)]dx=limλaλ[f(x)+g(x)]dx=limλaλ

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