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 continuous on \((0,\infty )\)and \(\int_1^\infty f (x)dx\)is convergent, then\(\int_0^\infty f (x)dx\)is convergent.

Short answer

True.

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).

Observation

It is given that\(f\)is continuous on\((0,\infty )\), then\(\int_0^1 f (x)dx\)is finite.

i.e;\(\int_0^1 f (x)dx\)is convergent.

Since\(\int_1^\infty f (x)dx\)is convergent, then\(\int_0^\infty f (x)dx\)can be written as

\(\int_0^\infty f (x)dx = \int_0^1 f (x)dx + \int_1^\infty f (x)dx\)

Since both\(\int_0^1 f (x)dx\)and\(\int_1^\infty f (x)dx\)are convergent, then\(\int_0^\infty f (x)dx\)is convergent.

So the statement is true.