Power series converges, Dirichlet test

Question

Let $a_n$ be a decreasing sequence of positive numbers with $\lim\limits_{n\to\infty}a_n=0$. Let $0<\delta<2$ be given, define $\Omega_\delta:=\{z\in\mathbb{C}\colon|z|\leq 1,|1-z|\geq\delta\}\neq\emptyset.$ Then the power series $$ \sum_{n=0}^\infty a_nz^n $$ converges uniformly for $z\in\Omega_\delta$.

I defined the partial sum $S_N(z)=\sum\limits_{n=0}^Na_nz^n=a_N\frac{1-z^{N+1}}{1-z}-\sum\limits_{n=0}^{N_1}\frac{1-z^{n+1}}{1-z}(a_{n+1}-a_n)$, with summation by parts. I also found that $|\frac{1-z^{n+1}}{1-z}|\leq \frac{2}{\delta}$ for all $z\in\Omega_{\delta}$ and all $n\in\mathbb{N}$. How can I show that $S_N$ converge uniformly? I read an answer concerning the Dirichlet test, but I don't see how this implies uniform convergence.

Answer 1

You can apply Dirichlet's test for uniform convergence, since

• the sequence $(a_n)_{n\in\mathbb N}$ is monotonic and converges to $0$;
• the sequence $\left(\sum_{n=0}^N z^n\right)_{N\in\mathbb N}$ is uniformly bounded (as you proved).