How can I show that the following is a martingale? Let $S_n = X_1 + ... + X_n$, where $X_1, X_2, ...$ are iid with $X_1 \sim \operatorname{Exp}(1)$.
I'd like to verify that
$$
 M_n = \frac{n!}{(1+S_n)^{n+1}} \cdot e^{S_n}
$$
is a martingale, that is $\mathbb{E}(M_{n+1} \mid \mathcal{F}_n) = M_n$, where $\mathcal{F}_n$ is the natural filtration.
So far I've tried to 'separate' $X_{n+1}$ from $S_{n+1}$:
$$
 \mathbb{E}(M_{n+1} \mid \mathcal{F}_n) =
 \mathbb{E} \Big( \frac{(n+1)!}{(1+S_n+X_{n+1})^{n+2}} \cdot e^{S_n} e^{X_{n+1}} \mid \mathcal{F}_n\Big)
$$
But I don't see what to do with the denominator there.
Is this a correct way to start (and if yes, what should I do next?) or is there some other way to show that $M_n$ is a martingale?
 A: Fix $n \in \mathbb{N}$. By the pull-out property, we have
$$\mathbb{E}(M_{n+1} \mid \mathcal{F}_n) = e^{S_n} (n+1)! \mathbb{E} \left( \frac{1}{(1+S_n+X_{n+1})^{n+2}} e^{X_{n+1}} \mid \mathcal{F}_n \right). \tag{1}$$
Consequently, it remains to calculate
$$\mathbb{E} \left( \frac{1}{(1+S_n+X_{n+1})^{n+2}} e^{X_{n+1}} \mid \mathcal{F}_n \right).$$
Since the random variables $(X_k)_{k \in \mathbb{N}}$ are independent, we know that $X_{n+1}$ and $\mathcal{F}_n$ are independent. On the other hand, $S_n$ is $\mathcal{F}_n$-measurable and so
$$\mathbb{E} \left( \frac{1}{(1+S_n+X_{n+1})^{n+2}} e^{X_{n+1}} \mid \mathcal{F}_n \right) = f(S_n) \tag{2}$$
where
$$f(s) := \mathbb{E} \left( \frac{1}{(1+s+X_{n+1})^{n+2}} e^{X_{n+1}} \right), \qquad s \geq 0.$$
Using that $X_{n+1}$ is exponentially distributed, we find
$$\begin{align*} f(s) &= \int_0^{\infty} \frac{1}{(1+s+x)^{n+2}} e^x \, \mathbb{P}_{X_{n+1}}(dx) \\ &= \int_0^{\infty} \frac{1}{(1+s+x)^{n+2}} \, dx. \end{align*}$$
This integral can be easily calculated. Plugging the result into $(2)$ and using $(1)$, this proves that $(M_n)_{n \in \mathbb{N}}$ is a martingale.
