This question arose whilst working on this problem.
Let $\ \left[{n\atop k}\right]\ $ denote the $\ (n,k)-$th Stirling number of the first kind.
Define $$ f(n) = \left[{n\atop 1}\right] r\ +\ \left[{n\atop 2}\right] r^2\ +\ \ldots\ + \left[{n\atop n}\right] r^n\ $$
I would like to find all the values of $\ r>0\ $ such that
$$ n \frac{f(n-1)}{f(n)} \to 1\quad \text{as}\ n\to\infty. $$
This is clearly true for $\ r=1,\ $ because $$ f(n) = \left[{n\atop 1}\right]\ + \left[{n\atop 2}\right]\ +\ \ldots\ +\ \left[{n\atop n}\right]\ = n!. $$
But I don't think I know enough about Stirling numbers of the first kind to figure out what happens when $\ r\neq 1.$
As pointed out in the comments, $\ f(n) = \prod_{k=0}^{n-1} (k+r).\ $ Thus,
$$ n \frac{f(n-1)}{f(n)} = \frac{n}{(n-1)+r} \to 1\quad \text{as}\ n\to\infty, $$
and this is true for any $\ r>0,\ $ or any $\ r<0: r\not\in\mathbb{Z}.$