What would be the radius of convergence of $\sum\limits_{n=0}^{\infty} z^{3^n}$? I know how to find the radius of convergence of a power series $\sum\limits_{n=0}^{\infty} a_nz^n$, but how does this apply to the power series $\sum\limits_{n=0}^{\infty} z^{3^n}$? Would the coefficients $a_n=1$, so that one may apply D'Alembert's ratio test to determine the radius of convergence? I would appreciate any input that would be helpful.

In a nutshell, my advice would be to forget that this is a power series and that you learned something called the ratio test, and to remember that a series $\sum\limits_nx_n$ cannot converge unless $x_n\to0$. In your case, $x_n$ is a power of $z$ hence $(x_n)$ does not converge to zero if $|z|\geqslant1$. In the other direction, a simple comparison such as $|x_n|\leqslant|z|^n$ for every $|z|\leqslant1$ yields the result.
And you might wish to read this.
