The following is a problem posed by Lerch in 1885.
Problem: For $m \in \mathbb{Z}$ and $p = 0 , 1 , 2 , \ldots , \left(m ! - 1\right)$. Show that
\begin{align} \lim_{z \to \exp \left(\frac{2 \pi i p}{m !}\right)} \sum_{n = 0}^{\infty} {z}^{n !} = \infty \\ \end{align}
where the limit $z \to \exp \left(\frac{2 \pi i p}{m !}\right)$ is taken along a radial line (with the argument $\frac{2 \pi i p}{m !}$) through that point.
Thoughts: Would I need to show that $m !$ and $n !$ line up just right so that all the powers of $z$ constructively superpose and diverge to $\infty$? Would I have to quote a result from number theory?