It is well-known that a power series sums to a function that is analytic at every point inside its circle of convergence and that conversely, if a function is analytic on an open disc then its Taylor (power) series converges throughout the disc.
Since analyticity is a property defined over open sets, does this mean that an analytic function can never have a power series that has only a point convergence (as opposed to a disc of convergence)?
Let $f(x)= \sum\limits_{n=0}^\infty n!(2x+1)^n$, which is convergent only at -0.5. Does this mean that $f$ is not analytic???