Confused on how the MGF of the exponential distribution works

Question

How can the MGF of the exponential dist with parameter $\lambda$ be $E[X^n] = \frac{n!}{\lambda^n}$ if the first and second moments are $\frac{1}{\lambda}$ and $\frac{1}{\lambda^2}$?

Where is the n! going that we get a one on top?

Answer 1

Let $M(t)$ be the moment generating function of a random variable. The power series at $t=1$ is $M(1)=\sum_0^\infty \frac{m_n}{n!}$ where $m_n$ is the $n^{th}$ moment. I hope this clarifies your concern.