I have a question about one of the solutions to an exercise from Serge Lang's Introduction to Complex Analysis. The context of the question is
and the question, including the given answer is
My Question: Why does it converge uniformly on $s \leq |z| \leq S$ rather than on $r \leq |z| \leq R$?
I understand how the uniform convergence implies the term by term differentiation but I do not understand why it converges uniformly on a slightly smaller ananullus rather than the one it is defined on.
Thank you in advance.
Because, for instance, the Laurent series of $\frac1{1-z}$ on $D_1(0)$ is$$\sum_{n=0}^\infty z^n,$$which converges uniformly on any closed disk $\overline{D_r(0)}$ when $r<1$, but not on $D_1(0)$. Also, the Laurent series of $\frac1{1-z}$ on $\{z\in\Bbb C\mid|z|>1\}$ is$$-\sum_{n=-\infty}^{-1}z_n,$$which converges uniformly on any annulus $\{z\in\Bbb C\mid r\leqslant|z|\leqslant R\}$ with $1<r<R$, but not on the annulus $\{z\in\Bbb C|1<|z|<\infty\}$.