What is the radius of convergence of the series $$\sum_{n=1}^{\infty} \frac{z^{n^{3}}}{n^4}?$$
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The sum converges when $z=1$. When $|z|>1$ we have $$\lim_{n\to \infty} \frac{z^{n^{3}}}{n^4} \neq 0.$$ Is this enough to conclude that the radius of convergence is 1?
Observe \begin{align} \sum^\infty_{n=1} \frac{z^{n^3}}{n^4} = \sum^\infty_{k=1}a_kz^k \end{align} where \begin{align} a_k = \begin{cases} n^{-4} & \text{ if }\ \ k=n^3\\ 0 & \text{otherwise}. \end{cases} \end{align} Then it follows \begin{align} \frac{1}{R} = \limsup_{k\rightarrow \infty} \sqrt[k]{|a_k|} = \lim_{n\rightarrow \infty} \frac{1}{\sqrt[n^3]{n^4}} = 1. \end{align}