Exponential Limit

Question

I do not know if this question has been asked before on this site, but I couldn't find it. I've been trying to approach it in a few different ways however none seems to lead to any concrete result. The problem is the following, reduced from moment generating functions for the gamma distribution and need to apply the Central Limit theorem to it, however it reduces to:

$\lim_{n\to\infty} {e^{-t\sqrt{n}}\left ( 1-\frac{t}{\sqrt{n}}\right)^{-n}} = e^{\frac{1}{2}t^2}$

It is pointless for me to provide my trials as they led to nothing (not even a useful intermediate step). So a full solution would be appreciated.

Answer 1

We have that

$$\left( 1-\frac{t}{\sqrt{n}} \right)^{-n}=e^{-n\log \left( 1-\frac{t}{\sqrt{n}} \right) }= e^{-n\left(-\frac{t}{\sqrt{n}}-\frac{t^2}{2n}+o(1/n)\right) }=e^{\left(t\sqrt{n}+\frac12t^2+o(1)\right)}=e^{t\sqrt{n}}e^{\left(\frac12t^2+o(1)\right)}$$

and therefore

$${e^{-t\sqrt{n}}\left ( 1-\frac{t}{\sqrt{n}}\right)^{-n}} = e^{-t\sqrt{n}}e^{t\sqrt{n}}e^{\left(\frac12t^2+o(1)\right)}=e^{\left(\frac12t^2+o(1)\right)}\to e^{\frac{1}{2}t^2}$$

Answer 2

Do a series expansion at ($n = \infty$), this gives you $$ e^{-t \sqrt{n}}\left(1-\frac{t}{\sqrt{n}}\right)^{-n} = e^{t^2/2} + \frac{1}{3\sqrt{n}}e^{t^2/2}t^3 + O(\frac{1}{n}) \,\,. $$ As $n \to \infty$, only the first term $e^{t^2/2}$ remains.