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???
Best Answer
The pointwise definition of an analytic function goes something like this: $f$ is said to be analytic at $x_0$ if it's differentiable $\infty$ times at $x_0$ and its Taylor series, centered at $x_0$, converges in some (maybe very small) complex disk centered at $x_0$.
A power series with a $0$ radius of convergence is not very interesting, since it cannot be used to define a function at more than one point. A function defined only at one point is not said to be analytic (see the above definition).