How to prove that the series $$\sum_{n=1}^{\infty} z^{n !}=z^{1!}+z^{2!}+z^{3!}+ \cdots $$ has the natural boundary $|z|=1$.
I did a similar problem to this where I was able to get an iterative functional relationship and show that all of them have singularities at $|z|=1.$ But I couldn't able to do it here. Any help in solving this is much appreciated.
Hint: Consider the behavior of this function along the radius terminating at $e^{i2\pi q},$ where $q\in \mathbb Q.$