solving power series expression?

Question

Can anyone show how to simplify $$\frac{\sum_{n=1}^\infty n\frac{n^{n-1}}{n!}x^{n-1}}{(-1+\sum_{n=1}^\infty \frac{n^{n-1}}{n!}x^n)^2}$$

Answer 1

small hint

the derivative of $$\frac{1}{1-f}$$ is

$$\frac{f'}{(-1+f)^2}.$$

Answer 2

Hint: The Lambert W function admits the expansion \begin{align*} \sum_{n=1}^\infty n^{n-1}\frac{x^n}{n!}=-W(-x)\qquad\qquad |x|<\frac{1}{e} \end{align*}