Problem from Remmert’s book chapter 4, 2a) p.122
I’m trying to solve the following problem
a) Let $a_n$ be real and decrease to 0. Show that the power series $\sum_{n \geq 0} a_n z^n$ converges compactly in $\mathbb{E} \setminus {1}$ Hint: investigate $(1-z)*\sum_{n \geq 0} a_n z^n$
b) Show that the logarithmic series $\sum_{n \geq 1} \frac{(-1)^{n-1}}{n} z^n$ converges compactly in $\mathbb{E} \setminus {1}$
Can anybody give me another hint? Unfortunately I’m stuck with nothing..
Let $K$ be any compact set in $\mathbb{E}\setminus \{1\}$. For any $z \in K$ we have $|z| \leqslant 1$ and $|z - 1| > \delta$ for some $\delta > 0$ -- since $\{1\} \not\in \bar{K}.$
Hence, for all $n \in \mathbb{N}$ and all $z \in K$ we have uniformly bounded partial sums
$$\left| \sum_{k=1}^n z^k\right| = \left|\frac{z - z^{n+1}}{1-z} \right| \leqslant \frac{|z|(1+ |z|^n)}{|1-z|} \leqslant \frac{2}{\delta}$$
Since $a_n$ converges to $0$ monotonically and uniformly (since there is no dependence on $z$) it follows by the Dirichlet test that $\sum_{n \geqslant 0} a_n z^n$converges uniformly on the compact set $K$.