Radius of convergence of $\sum_{n\ge1}z^{n^4}$

Question

What is the radius of convergence of $$\sum_{n\ge1}z^{n^4}$$

The exponent $n^4$ is troublesome. Couldn't solve it.

Answer 1

HINT: For every 0-1 sequence with infinite number of 1's we have $$ \limsup\sqrt[n]{a_n}=1. $$

Remark: In fact, this gives radius 1 for any nonnegative, bounded sequence with infinite number of positive terms.

Answer 2

If $|z|<1$, then $|z^{n^4}|=|z|^{n^4} \le |z|^n$, hence the power series converges for $|z|<1$.

The power series diverges for $z=1$. Hence the radius of convergence $=1$.