Let $S$ be either $\Bbb R$ or $\Bbb C$. Given any sequence $(a_n)$, with each $a_n\in S$, we may form a function $f(x)=\sum_{n=0}^\infty a_nx^n$ whose domain is the set of points $x\in S$ such that the series is absolutely convergent, and then take $F$ to be the restriction of $f$ to the interior of its domain.
The question is: Is $F$ $S$-analytic? That is, is it the case that $F$ has a Taylor series expansion and matches its Taylor series in some neighborhood of every point in its domain?
Clearly $f$ need not be, because there are sequences such as $a_n=n!$ which converge only at $0$, so that the domain of $f$ is $\{0\}$ which is not open, and this is a necessary condition of analyticity. We can resolve this problem by considering the interior of the domain so we get an open set, which is why $F$ is defined as such. $F$ is obviously expandable as a Taylor series at $0$ (assuming $0\in\mbox{dom }F$) by definition, but at other points it is not clear what properties hold.