I wanted to know how to derive the series for ln(2)

I did pretty much the same thing. Integrating both 1/(1+x) and the infinite geometric series term by term. The bottom line is that there is convergence to ln(2) when x = 1 but someone else is going to have to weigh in as to how the radius of convergence -1<x<1 in the original sequence now somehow changes to -1<x<=1 for ln(x+1). Not much of a help for you but I agree with you that you have identified the key sticking point.