Can the function $f(z) = \sum_{n = 0}^\infty z^{n!}$ can be analytically extended outside the circle $|z|=1$
My attempt : I was trying using some some radius convergence and analytic continuation of a function along a path. But nothing find effective.Any help/hint in this regards would be highly appreciated. Thanks in advance!
If $\omega\in\mathbb C$ is a root of unity, then $n\gg1\implies\omega^{n!}=1$. Therefore,$$\lim_{\lambda\to1^-}\bigl|f(\lambda\omega)\bigr|=+\infty.$$So, since the roots of unity are dense in $S^1$, no, you cannot extend it analitically to a connected region strictly containing $D(0,1)$.