Question

Determine whether the statement below is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement (a counterexample).

"If f(x)≤g(x) and ₀∫[infinity] g(x)dx diverges, then ₀∫[infinity] f(x)dx also diverges."
A.True
B.False

Answer 1

The statement is false and a counterexample can be given to disprove it.

Step-by-step explanation:

The statement is false and a counterexample can be given to disprove it.

Counterexample: Let's consider the functions f(x) = 1/x and g(x) = 1/(x^2).

Clearly, f(x) ≤ g(x) for all x greater than 1.

Now, ∫1∞ g(x) dx = ∫1∞ 1/(x^2) dx. This integral converges as it is a p-series with p = 2 and p > 1.

However, ∫1∞ f(x) dx = ∫1∞ 1/x dx. This integral diverges as it is a p-series with p = 1 and p ≤ 1.

Therefore, the statement is false and the given counterexample disproves it.