Can this type of function, defined on the closed unit disk, be analytically continued into the complex plane?

Question

Let ${\Bbb D}:=\{z\in\Bbb C:|z|<1\}$ and $f:\overline{\Bbb D}\to\Bbb C$ be given by $f(z):=\sum_{n=0}^\infty a_nz^n$, where the constants $a_n\in\Bbb C$ are such that $\sum_{n=0}^\infty |a_n|$ is convergent but, for any given $x>1$, the quantity $|a_n|x^n$ is unbounded as $n\to\infty$. Can $f$ be analytically continued into the complex plane and, if so, what further conditions on the $a_n$ are known that will ensure this?

Answer 1

Your conditions imply that the convergence radius of the power series of $f$ centered at $0$ is $1$. This means that $f$ cannot be analytically extended to a full neighborhood of $\overline{\mathbb{D}}$.

There are examples of functions that cannot be analytically continued to any larger set than $\overline{\mathbb{D}}$. See this Mathoverflow answer for examples. My favorite one given there is

$$ f(z) = \sum_{n=1}^\infty \frac{z^{n!}}{n^2}. $$