Hey I have a short question about region of convergence and the region of uniform convergence.
Let $G$ be the region of convergence of a function $f\left(z\right)$. Note that $f\left(z\right)$ is described by some serie.
My question is, is it possible that the region of uniformly convergence of the function $f\left(z\right)$ is smaller (is within) the region of convergence of $f\left(z\right)$?
A complex power series $\sum_{k=0}^\infty a_k (z-p)^k$ must either converge only at $p$, converge pointwise on a disk $|z-p| < r$ and diverge when $|z-p| > r$, or converge for all $z$. Pointwise convergence on the boundary must be decided on a case by case basis.
The proof is via comparison of the series of absolute values to a geometric series. This argument also gives uniform convergence on any closed disk $|z-p| \leq r' < r$. This argument can be found in any text on complex analysis.