I'm trying to solve a improper integral exercise. It follows:
Using the comparison test determine whether the following integral converges or diverges
$$\displaystyle \int_{1}^{\infty} x^3 \cdot e^{-x} \ dx$$
Given that the answer is convergent, I assume I need to compare it to a function that is greater than the given one (that will allow me to conclude that, if the new one is convergent, the given one will also be convergent). I am having a hard time coming up with a function to compare the given one with. Using for example $g(x) = x^3$ would definitely not work (as it is divergent).
Any help is highly appreciated.