Large $n$ asymptotic of $\int_0^\infty \left( 1 + x/n\right)^{n-1} \exp(-x) \, \mathrm{d} x$ While thinking of 71432, I encountered the following integral:
$$
  \mathcal{I}_n  = \int_0^\infty \left( 1 + \frac{x}{n}\right)^{n-1} \mathrm{e}^{-x} \, \mathrm{d} x
$$
Eric's answer to the linked question implies that $\mathcal{I}_n \sim \sqrt{\frac{\pi n}{2}} + O(1)$.
How would one arrive at this asymptotic from the integral representation, without reducing the problem back to the sum ([added] i.e. expanding $(1+x/n)^{n-1}$ into series and integrating term-wise, reducing the problem back to the sum solve by Eric) ? 
Thanks for reading. 

Interesting. I've got a representation
$$
\mathcal{I}_n  = n e^n \int_1^\infty t^{n-1} e^{- nt}\, dt
$$
which can be obtained from yours by the change of variables $t=1+\frac xn$. After some fiddling one can get
$$
2\mathcal{I}_n= n e^n \int_0^\infty t^{n-1} e^{- nt}\, dt+o(\mathcal{I}_n)=
n^{-n} e^n \Gamma(n+1)+\ldots=\sqrt{2\pi n}+\ldots.
$$
