Expectation property of moment generating function

Question

Let $X$ be a non-negative random variable and define $M_X(t) := \mathbb{E}[e^{tX}]$. Suppose that $M_X(t) < \infty \mbox{ for all } t < \alpha$, where $\alpha > 0$.

It can be shown that this condition implies $\mathbb{E}[X^k]<\infty$ for all $k$, i.e. all moments exist.

How can we establish that $\mathbb{E}[X^ke^{tX}] < \infty$ for all $k$ and $ t < \alpha$?

Answer 1

Let $t<t'<\alpha$. Then $e^{(t'-t)x} \geq \frac {(t'-t)^{k}x^{k}} {k!}$ so $x^{k}e^{tx} \leq \frac {e^{t'x} k!} {(t'-t)^{k}}$. Hence $EX^{k}e^{tX}\leq M(t') \frac { k!} {(t'-t)^{k}}<\infty$.