Power series without analytic continuation Given a formal power series $\sum a_n z^n$ and a radius of convergence $R>0$, there are various ways to extend the function to the boundary such as


*

*Abel's theorem

*Fatou's lemma 

*$H^\infty$ theorem.


What is an example of a function that does (almost) nowhere to the boundary? Which power series are proven to not possess an analytic continuation beyond the radius of convergence.
The craziest things what I can construct is a finite number of essential singularities using an entire function $f$, which is not a polynomial, and looking at something like $z \mapsto f(1/z)$.

The sum $f(z) = \sum_{n = 0}^\infty z^{n!}.$ extends nowhere outside of $|z|\lt1$; there are infinitely many points in the set of n-th roots of unity in the boundary each of which blow up to infinity; the set of n-th roots is dense in the boundary $|z|=1$, so the series cannot squeeze out anywhere.  
