$\int_{1}^{\infty} \frac{p(x)}{e^x} dx$ converges?

Question

How to decide whether the improper integral $\int_{1}^{\infty} \frac{p(x)}{e^x} dx$ converges or diverges, when $p(x) \in \mathbb R[x]$?

Answer 1

Hint: Use integration by parts and obtain \begin{eqnarray} \int_{1}^{\infty} \frac{p}{e^x} dx &=& \int_{1}^{\infty} pe^{-x}\,dx\\ &=& \dfrac{(p+p'+p''+p'''+\cdots)(1)}{e} \end{eqnarray}