Take some power series $f(z) = \sum a_n z^n$ with a finite non-zero radius of convergence. I can rearrange the terms of the series, say, to get a different infinite series $$f_{\sigma}(z) = \sum_{n=0}^{\infty} a_{\sigma(n)} z^{\sigma(n)}$$ where $\sigma$ is a bijection from the nonnegative integers $\mathbb{N}$ to $\mathbb{N}$. Doing this rearrangement doesn't change anything in the interior of the disk of convergence, since the series converges absolutely inside the disk.

But suppose the original power series had conditional convergence at some points on the boundary of the disk. (The series $$f(z) = \sum_{n=1}^{\infty} \frac{z^n}{n}$$ is one example.) What is the space of possible functions $f_{\sigma}$ that could result from rearrangements? I'm thinking that we would have something at least vaguely akin to Riemann's rearrangement theorem, but that theorem, as far as I know it, can only deal with series of numbers, not series of functions.