The sequence of polynomial equations $$ (a+n)x^n+(b-n)x^{n-1}+f(n)=0 $$
has real roots near 1 having an asymptotic series $$ x_n=1+\frac{y_n-a-b}{n}+O(y_n^2n^{-2}), $$ where $$ y_n=W(-e^{a+b}f(n)). $$ Here $W$ is the Lambert function.
Now, I want to apply this result to the special sequence of polynomials $$ x^n-x^{n-1}-1=0. $$ So I guess I have to choose $$ a=1-n,\qquad b=n-1,\qquad f(n)=-1. $$
Then I should have by above $$ x_n=1+\frac{y_n}{n}+O(y_n^2n^{-2}) $$ with $y_n=W(1)$.
On the other hand, I know that $$ x_n\sim1+\frac{\log n}{n}. $$
Do these two results fit together? Maybe $y_n=W(1)$ is equivalent to $y_n=\log(n)$?