determine the maximal open subset $G\subset\mathbb{C}^2$ such that the series $f$ converges in $\mathcal{O}(G)$ in the topology of Frechet space.

Question

Consider $f(z_1,z_2)=\sum\limits_{j=0}^\infty(z_1+z_2)^j$,determine the maximal open subset $G\subset\mathbb{C}^2$ such that the series $f$ converges in $\mathcal{O}(G)$ in the topology of Frechet space.
I guess $G=\{|z_1|+|z_2|\le 1\}$.For any region $U\supsetneqq A$,$\exists (z_1,z_2)$ such that $|z_1|+|z_2|=b>1$,i.e.$z_1+z_2=be^{i\theta}$ for some $\theta\in\mathbb{R}$.I don't know how to show $\sum\limits_{j=0}^\infty b^je^{ji\theta}$ diverges for any $\theta$.I want to rotate the coordinate axis to guarantee $\theta=0$.But I think it is unreasonable after a second thought.Can you help me explain why $\sum\limits_{j=0}^\infty b^je^{ji\theta}$ diverges for any $\theta$?Thanks for advance!

Answer 1

You must not confuse the domain of convergence of the Taylor series of $f$ with the domain where the given series converges locally uniformly.

You are given a specific series, let's write $g_j(z_1,z_2) = (z_1+z_2)^j$, and you have to find the maximal open subset $G \subset \mathbb{C}^2$ where the series

$$\sum_{j=0}^\infty g_j(z_1,z_2)$$

converges locally uniformly. That is in this case a strictly larger domain than the domain of convergence of the Taylor series of $f$. Weierstraß' $M$-test should help you find $G$ - or at least proving that it is the largest domain where the series converges locally uniformly once you found the right $G$.