Question

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\((1,\infty )\)and\(\mathop {\lim }\limits_{x \to \infty } f(x) = 0\), then\(\int_1^\infty f (x)dx\)is convergent.

Short answer

False

Step-by-step solution

Definition

Integrals is convergent if the associated limit exists and is a finite number (i.e. it's not plus or minus infinity).

Example

We’ll disprove this statement by contradictory example as this is false statement.

Let us consider\(f(x) = \frac{1}{x}\)

Clearly here\(f\)is a continuous and decreasing function on\((1,\infty )\)and\(\mathop {\lim }\limits_{x \to \infty } f(x) = 0\)

That is, \(\mathop {\lim }\limits_{x \to \infty } \frac{1}{x} = 0\)

But \(\int_1^\infty {\frac{1}{x}} dx\)is divergent.

So the statement is false.