$\sum_{n=0}^{\infty} a_n x^n$ and $\sum_{n=0}^{\infty} a_{n^2} x^n$ with different radii of convergence

Question

Could you give an example of $$\sum_{n=0}^{\infty} a_n x^n$$ and $$\sum_{n=0}^{\infty} a_{n^2} x^n$$ that have different radii of convergence?

Answer 1

The simplest example (pointed out by David Mitra) probably is: $$ a_n=\left\{\begin{array}{cl} 0,& n=m^2, \, m\in \mathbb{N} \\ 1, &\text{otherwise} \end{array} \right. $$

$\sum a_nx^n$ has a radius of convergence $1$ while $\sum a_{n^2}x^n$ converges for all $x\in \mathbb{R}$.