consider the rational function :

We wish to find an expansion in z that is valid for all x and z. a Bernoulli-type expansion is only valid for :

Therefore, we consider an expansion around z=1 of the form :

Where are suitable functions in x that make the expansion converge. the first two are given by :

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 !?