Can a uniform convergent power series be invariant under rearrangement

Question

Suppose $S$ is a power series, uniformly convergent. Now under rearrangement, does the sum stays the same? I know, if the series is absolute convergence then this is true, but under uniform convergence does it holds? I am assuming, the sum always converges, as we taking the domain to be the radius of convergence.

Answer 1

Assume wlog the power series centered at $0$ (otherwise translate); then convergence at only one point $|z_0|=r>0$ implies absolute convergence on the disc $|z|<r$, so the question is kind of moot except for the boundary disc $|z|=r$ where the result is not true.

One can have a power series with radius $1$, uniformly convergent on $|z|=1$ but not absolutely convergent there (eg $\sum_{n \ge 1}\frac{e^{in \log n}}{n}z^n, |z| \le 1$ or for real coefficients just take the real part so $\sum_{n \ge 1}\frac{\cos (n \log n)}{n}z^n, |z| \le 1$).

A convergent but non absolutely convergent series of complex numbers cannot be arbitrarily rearranged by the usual Riemann rearrangement theorem applied to the real or imaginary parts as one those at least cannot be absolutely convergent

(this last part is independent of any uniform convergence conditions or the fact the series is a power series as it applies to any point where the series is convergent but not absolutely convergent - the non trivial part is the existence of uniformly convergent power series that are not absolutely convergent on the boundary circle of convergence).