Radius of convergence problem

[mmzaj]

consider the rational function :

$$f(x,z)=\frac{z}{x^{z}-1}$$
$$x\in \mathbb{R}^{+}$$
$$z\in \mathbb{C}$$

We wish to find an expansion in z that is valid for all x and z. a Bernoulli-type expansion is only valid for :
$$\left | z\ln x \right |<2\pi$$
Therefore, we consider an expansion around z=1 of the form :
$$\frac{z}{x^{z}-1}=\sum_{n=0}^{\infty}f_{n}(x)(z-1)^{n}$$
Where $f_{n}(x)$ are suitable functions in x that make the expansion converge. the first two are given by :
$$f_{0}(x)=\frac{1}{x-1}$$

$$f_{1}(x)=\frac{x-x\ln x -1}{(x-1)^{2}}$$
now i have two questions :
1-in the literature, is there a similar treatment to this specific problem !? and under what name !?
2- how can we find the radius of convergence for such an expansion !?

 

[mmzaj]

it's not so hard to prove that the functions $f_{n}(x)$ have the general form :
$$f_{n}(x)=\frac{(-\ln x)^{n-1}}{n!}\left(n\text{Li}_{1-n}(x^{-1})-\ln x\;\text{Li}_{-n}(x^{-1})\right)$$