Let $f(z)=\sum_{j=0}^{\infty}z^j$ for $|z|<1$. For what values of $\alpha$ ($|\alpha|<1$) does the Taylor expansion of f(z) about $z=\alpha$ yield a direct analytic continuaton of f(z) to a disk extending outside $|z|<1$?
I thought that the answer would be $\alpha=0$ since then the function f(z) would be constant. And like a zero degree polynomial it would be analytic in the entire complex plane and thus have extended the function to be total. But the answer key says $-1\leq\alpha<0$, what is wrong with my reasoning?