Let $f(x) = \sum_{n=0}^{\infty} x^n$ for $|x| < 1$ (so $f(x) = \frac{1}{1-x}$). If $|g(x)| < 1$ for all $|x| < r$, what is the radius of convergence of the composed series $f(g(x)) = \sum_{n=0}^{\infty} g(x)^n$?