Show that the following series converge on $\mathbb{C}$:
$\sum_\limits{n=1}^{\infty}\dfrac{z^n}{a^{\sqrt{n}}}$ where $z\in\mathbb{C}$ is a variable and $a>0$.
I f I apply the root test I get $\lim_{n\to\infty}|z|a^{-\frac{1}{\sqrt{n}}}=|z|$ so it converges if $|z|<1$. However it was pointed out that if $a<1$ the series do not converge.
Question:
How can I prove that the series does not converge for $a<1$? For $a<1$, $\lim_{n\to\infty}|z|a^{-\frac{1}{\sqrt{n}}}=|z|$ still holds?