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 \(\int_a^\infty f (x)dx\)and \(\int_a^\infty g (x)dx\)are both convergent, then \(\int\limits_a^\infty {(f(x) + g(x))} dx\)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).

Explanation

Given that \(\int_a^\infty f (x)dx\)and \(\int_a^\infty g (x)dx\)are both convergent.

\( \Rightarrow \int_a^\infty g (x)dx = \mathop {\lim }\limits_{t \to \infty } \int\limits_a^t {g(x)dx} \)and\(\int_a^\infty {f(x)} dx = \mathop {\lim }\limits_{t \to \infty } \int\limits_a^t {f(x)dx} \)

Then \(\int\limits_a^\infty {(f(x) + g(x))} dx\) can be expressed as:

\(\begin{array}{c}\int\limits_a^\infty {(f(x) + g(x))} dx = \mathop {\lim }\limits_{t \to \infty } (\int\limits_a^t {f(x)dx} + \int\limits_a^t {g(x)dx} )\\ = \mathop {\lim }\limits_{t \to \infty } \int\limits_a^t {f(x)dx} + \mathop {\lim }\limits_{t \to \infty } \int\limits_a^t {g(x)dx} \,\,\,(As\,both\,\lim its\,are\,finite)\\ = \int\limits_a^\infty {(f(x))} dx + \int\limits_a^\infty {(g(x))} dx\end{array}\)

As given both limits are convergent, therefore, \(\int\limits_a^\infty {(f(x) + g(x))} dx\)is also convergent.

So the given statement is true.