Domain of Convergence of Taylor Series in Several Variables

Question

I have the following function in 2 variables:

$$\frac{1}{1+z_1+2z_2}$$

I have to find it's Domain of Convergence. My initial thought was that it'll simply be $|z_1 + 2z_2|<1$, but then I realised, following a standard result, the domain of convergence is supposed by invariant under $\mathbb{T}^n$ action, $|z_1 + 2z_2|<1$ isn't invariant.

Also the following expansion of the series doesn't help either$$\sum_{k=0}^{\infty}(-1)^k (z_1 + 2z_2)^k$$

Edit: I have a guess now, it's $|z_1|+2|z_2|<1$. I just have to proof this is the domain.

For that, one way, is to produce $\theta_1, \theta_2$ such that for $|z_1|+2|z_2| \geq 1$, $\sum (e^{i\theta_1}z_1 + 2e^{i\theta_2}z_2)^k$ diverges.

Any suggestions?

Answer 1

We get an answer for $\theta_1 = -\text{Arg}(z_1),\ \theta_2 = -\text{Arg}(z_2)$