Radius of convergence of a power series whose coeffecients are "discontinuous" I have a power series:
$s(x)=\sum_0^\infty a_n x^n$
with
$a_n=
\begin{cases}
1,  & \text{if $n$ is a square number} \\
0, & \text{otherwise}
\end{cases}$
What is the radius of convergence of this series?
I tried to use ratio test, but as$\lim\limits_{n \to \infty} \frac{a_n x^n}{a_{n+1} x^{n+1}}$does not exist, I don't know how to apply the ratio test on the problem.
Thanks in advance!

Although $\lim_{n\to\infty}\frac{a_{n+1}}{a_n}$ and $\lim_{n\to\infty}\sqrt[n]{a_n}$ don't exist, you can still find $\limsup_{n\to\infty} \sqrt[n]{a_n}$. Since
$$
\sup_{m\ge n} \sqrt[m]{a_m}=1
$$
for all $n\in\mathbb{N}$, $\limsup_{n\to\infty}\sqrt[n]{a_n}=1$ and so the radius of convergence is $1$.
