finding the radius of convergence of a complex power series

Question

I am trying to find radius of convergence of

$$ \sum_{n=0}^{\infty} z^{a^n} $$

where $a>1$ integer.

I obviously want to use $1/R = \limsup ( |c_n| )^{1/n}$. Is there a way to write $z^{a^n}$ in the form $c_nz^n $?

Answer 1

Let

$b_n = \begin{cases}1, &n \text{ is a power of } a;\\ 0, &\text{otherwise.} \end{cases}$

Then, $\sum{z^{a^n}}=\sum{b_nz^n}$, note that there are infinity many $1$s in the serie of $b_n$, then $\lim \sup |b_n|^{1/n}=1^{1/n}=1$ . That is $R=1$.

Answer 2

Hint: Let $$b_n = \begin{cases}1, &n \text{ is a power of } a;\\ 0, &\text{otherwise.} \end{cases}$$

Then the series is $\sum_{n} b_n z^n$. What is $\limsup |b_n|^{1/n}$?