Why is this random variable integrable?

Question

Suppose to have $X$ a random variable a.s. positive and let $t>0$. Why does it hold that the random variable $$X^ne^{-tX}$$ is integrable, for $n>1$?

For $n=1$ I have that $E[X e^{-tX}] \leq E[X \frac{1}{tX} ]= \frac{1}{t}$ and so is integrable.

But what about the other cases? How can I show it?

Answer 1

If $x \ge0 $ then $x^n e^{-tx} = {x_n \over e^{tx}} \le {x^n \over 1+ {(tx)^n \over n!} } \le {n! \over t^n}$. Hence $E[X^n e^{-tX}] \le {n! \over t^n}$.