Limit comparison test:
If the positive functions f and g are continuous on $$\left [ a, \infty \right )$$ and if $$\lim_{x\rightarrow \infty }\frac{f(x)}{g(x)}=L$$ where $$0< L< \infty $$ then $$\int_{a}^{\infty }f(x) \ and \ \int_{a}^{\infty }g(x)$$ both converge or both diverge.
Now let's assume the opposite. Under the same conditions, except $$L =\left \{ 0,\infty \right \}$$
Now, can we say one of $$\int_{a}^{\infty }f(x) \ and \ \int_{a}^{\infty }g(x)$$ converges and the other diverges?