Recall that the radius of convergence $R$ of the series $\sum\limits_na_nz^n$ is such that for every positive real number $r>R$, the real valued sequence $(x^{(r)}_n)$ defined by $x^{(r)}_n=|a_n|r^n$ is unbounded and for every positive real number $r, this same sequence $(x^{(r)}_n)$ is bounded.
This dichotomy determines $R$ uniquely but of course much more is true since, for every $r, $x^{(r)}_n\to0$ exponentially fast. On the other hand, the possible behaviours of $(x^{(R)}_n)$ (that is, at the critical value $r=R$) are more diverse since one can observe anything between (non exponential) convergence to zero and (non exponential) unboundedness.
Application: Consider any complex valued sequence $(a_n)$ such that $|a_n|\in\{0,1\}$ for every $n$ and introduce the set of indices $N=\{n\mid|a_n|=1\}$. Then the radius of convergence of the series $\sum\limits_na_nz^n$ is $R=+\infty$ if $N$ is finite, and $R=1$ if $N$ is infinite.