For each $k\in\mathbb N$, I am trying to find $\lim_\limits{n\to\infty} a_n^n$ where $$a_n=\frac{(n!)^2}{(n+k)!(n-k)!}$$
Simplifying the factorials I have $$a_n=\frac{n(n-1)\ldots(n-k+1)}{(n+k)(n+k-1)\ldots(n+1)}$$
I tried to rewrite this as $$a_n=\frac{\left(1-\frac{1}{n}\right)\left(1-\frac{2}{n}\right)\ldots\left(1-\frac{k-1}{n}\right)}{\left(1+\frac{k}{n}\right)\left(1+\frac{k-1}{n}\right)\ldots\left(1+\frac{1}{n}\right)}$$
But all this tells me is that $a_n\to 1$.
Using Stirling's approximation, I think I also have $$a_n\sim \frac{n^{2n+1}}{(n+k)^{n+k+\frac{1}{2}}(n-k)^{n-k+\frac{1}{2}}}$$
Don't think this helps either. Any hint would be great.
Best Answer
Hint We know that $\text{lim}_{n\to \infty} (1+\frac{x}{n})^n=e^x$ can you see how to proceed.