The function $f(z) = \sum_{n=0}^\infty z^n$ is expanded into a Taylor series in a neighbourhood of the point a where $|a|< 1$. For what values of a the newer expansion provides the analytic continuation of $f(z)$ outside of the unit circle?
So my understanding of a is that it represents the centre in the circle of convergence for the Taylor series. So I need to choose values for a that recentres $f(z)$ outside of the unit disk.
If I let $a = 0$, then I will just have my original function, $f(z) = \sum_{n=0}^\infty z^n$ . So, will choosing $-1<a<0$ and $0<a<1$ ensure that I am outside of the unit circle?