How to show $\sum_{i=1}^{n-1} \frac{i(n-2)!}{(n-1-i)!n^{i+1}} \sim 1/n$

Question

How can one compute the large $n$ asymptotics of

$$\sum_{i=1}^{n-1} \frac{i(n-2)!}{(n-1-i)!n^{i+1}}\;?$$

My guess is that it is $1/n$ but I don't know how to show that.

Answer 1

HINT: use the fact that

$$\sum_{i=1}^{n-1} \frac{i(n-2)!}{(n-1-i)!n^{i+1}}=\int_0^{\infty} x(1+x)^{n-2} e^{-n x} \ dx$$

From the fact that $$\lim_{n\to\infty} n\int_0^{\infty} x(1+x)^{n-2} e^{-n x} \ dx=1$$

we conclude that for $n$ large enough we have

$$\sum_{i=1}^{n-1} \frac{i(n-2)!}{(n-1-i)!n^{i+1}}\sim \frac{1}{n}$$