Power series in several variables

Remarks

Convergence here is absolute convergence, since the index set $\N^n$ has no preferred order when $n \ge 2$. Example: for $n = 2$, $a = 0$ and $c_\alpha = 1$ for all $\alpha$, at $z = (1/2, 1/3)$ the sum of absolute values is $\sum_j 2^{-j} \sum_k 3^{-k} = 2 \cdot \tfrac32 = 3 < \infty$, so the series converges, with sum $3$. Non-example: the same series at $z = (1/2, 1)$ has the terms $2^{-j}$ with $k$ arbitrary, so the sum of absolute values is infinite and the series does not converge there. Grouping by total degree $\abs{\alpha} = \alpha_1 + \cdots + \alpha_n$ writes the series as $\sum_{m \ge 0} P_m(z - a)$, with $P_m$ a homogeneous polynomial of degree $m$. For $n = 1$ this is the ordinary power series, and in that case absolute convergence holds at every point inside the disc of convergence.