Yes, because any compact of D(0,1) is included in such a subdisc centered at 0 (distance to the boundary is positive).

The idea is to expand the ratio into a power series ( ) so as to get a series of powers (for some depending on ), and gather the terms with same power by re-organizing the summation to have the usual ...Subsequently, I must find the power series coefficients. How do I do this? Does anyone have general guidelines or a worked example I could analogize?