Answer
False
Work Step by Step
Consider $f(x)=\frac{1}{x}$ and $g(x)=-\frac{1}{x}$, then
$\int _{a}^{\infty}f(x)dx$ is divergent and
$\int_{a}^{\infty}g(x)$ is divergent, but $\int _{a}^{\infty}
(f(x)+g(x)) dx =\int _{a}^{\infty}0 dx=0$ is convergent.
Hence, the given statement is false.
