Why does this term converge to $1$?

Question

$$\frac{n!}{(n-k-i)!n^{k+i}}\rightarrow1$$ as $n\rightarrow\infty$.

I wonder why that is? Thanks for an explanation!

Answer 1

Let's look at one example: Suppose $i + k = 5$.

Then you have:

$$\frac{n!}{(n-5)!n^5} = \frac{n(n-1)(n-2)(n-3)(n-4)}{n^5}$$

As $n \rightarrow \infty$, the expression above will converge to $1$.

Picking other values of $i + k$ will lead you down a similar road; the task remaining for you is to write it up in a general form (which is essentially done in the comments already posted).