I want to show that i) $F_2(z)=\cfrac{1}{4}\pi i-\cfrac{1}{2}ln2+\left(\cfrac{z-i}{1-i}\right)+\cfrac{1}{2}\left(\cfrac{z-i}{1-i}\right)^2+\cfrac{1}{3}\left(\cfrac{z-i}{1-i}\right)^3+\ldots$ converges for $|z-i|<\sqrt{2}$ and ii) how can I show that $F_2(z)$ is an analytic continuation of the function $F_1(z)=z+\cfrac{1}{2}z^2+\cfrac{1}{3}z^3+\cfrac{1}{4}z^4+\ldots.$?
My first try was demostrate that $F_1(z)$ converges only if $|z|<1$ and later I found that, in general, the function $-ln(1-z)$ represents all the analytic continuations of the function $F_1(z)$. But I have problems with i) and ii).
Any help is welcome and appreciate!