Let $K$ be a fixed positive integer, show that $$\dfrac{d^i}{dx^i}\left(\frac{x}{\ln(1-x)}\right)^{1/K} \Bigg|_{x=0}>0,  ~~~\forall i\in N^{+}$$

The problem is from when I solve this:$$\left(\sum_{i=1}^{n}a_{i}x^i\right)^K=\dfrac{x}{\ln{(1-x)}},\text{ show that }a_{i}>0,\forall i\in N^{+}.$$ Maybe this  $$f(x)=\dfrac{x}{\ln{(1-x)}}$$ is special function? Is there a background to this conclusion?