Power series in $\mathbb{C}^2$.

Question

In complex analysis of one variable, we have the series $\sum_nz^n$ is convergent for $|z|<1$. If i consider the same in $\mathbb{C}^2$, that the series $\sum_{n_1,n_2}z_1^{n_1}z_2^{n_2}$, for $|z_i|<1$ $i=1,2.$ Is this series convergent? Or is this true in general spaces?

Answer 1

Hint:

$$ \sum_{n_1=0}^{N_1}\sum_{n_2=0}^{N_2} z_1 ^{n_1}z_2 ^{n_2} = \sum_{n_1=0}^{N_1}z_1 ^{n_1} \times \sum_{n_2=0}^{N_2} z_2 ^{n_2} $$