Find the Taylor expansion around z = 0 and the radius of convergence of .

I started out by simplify the expression. Furthermore in separated fashion, I don't know the mathematical term.

= = Expansion = = =

Questions

I couldn't find any formula giving the expansion like for instance . How would I go about finding it? Is there a golden rule telling me how far I should develop if it isn't stated? I just happened to chose the power of 4 because my table of formulas does. Is the convergence radius = 1, (z-1) i the closest singularity to z = 0, right?