Let's consider the series $$s=\sum_{n=0}^\infty \frac{a^{n^2}}{b^{cn}}$$ where $a,b\in\mathbb{C}-\{0\}$ ($\mathbb{C}^*$) and $c\in \mathbb{Z},c\neq 0$.
Which one would be a good approach to find the values of convergence of $s$?
I mean, I can consider the cases where $|a|>1$, where $|a|=1$, and $|a|<1$ (Each of these cases gets some subcases for conditions on $b$ and $c$), but it's too long an approach.
I think the convergence is mainly dependent on the value of $a$, and I think there should be a better way to check it.
HINT: $$ \frac{a^{n^2}}{b^{cn}}=\left(\frac{a^n}{b^c}\right)^n $$ thus your series converges if and only if $|a|^n<|b^c|=|e^{c\log b}|=|b|^c$ definitely, and this could happen iff $|a|<1$, for every $b$ and $c$ or if $|a|=1$ when $|b|^c>1$.