Uniform convergence of power series with decreasing coefficients

Question

Let $\{a_n\}$ be a monotone decreasing sequence of real numbers, with limit $0$.

Prove that, for each $0<\delta<2$, the power series

$\sum_{n=0}^{\infty} a_n z^n$

converges uniformly on each $\Omega_\delta = \{z : |z|\leq 1 , |z-1|\geq \delta\}$

I have no idea on how to deal with the points in the boundary. Thanks in advance!

Answer 1

Since $a_n \downarrow 0$ (trivially uniformly with respect to $z$) the Dirichlet test implies uniform convergence when $\left| \sum_{k=1}^n z^k\right|$ is uniformly bounded for all $z \in \Omega_\delta$ and for all $n$.

In this case,

$$\left| \sum_{k=1}^n z^k\right| = \left|\frac{z - z^{n+1}}{1-z} \right| \leqslant \frac{|z|(1+ |z|^n)}{|1-z|} \leqslant \frac{2}{\delta}$$