Ring of convergent power series

Remarks

The condition says that the series converges absolutely on the closed polydisc of radius $r$ about $0$; it is equivalent to convergence on some neighbourhood of $0$. Sums and products of convergent series are convergent, so $\C\{z_1, \dots, z_n\}$ is a subring of $\C[\![z_1, \dots, z_n]\!]$. Example: $f = \sum_{k \ge 0} z_1^k z_2^k$ is convergent: with $r = 1/2$, $\sum_k r^{2k} = 4/3 < \infty$. Non-example: $g = \sum_{k \ge 0} k!\, z_1^k$ is not convergent, since $k!\, r^k \to \infty$ for every $r > 0$. For $n = 0$ the ring is $\C$. A convergent series is a unit if and only if its constant term is nonzero, so the ring is local with maximal ideal $(z_1, \dots, z_n)$. Sending a series to its sum near $0$ gives an isomorphism with the ring $\mathcal{O}_{n,0}$ of germs of holomorphic functions at $0$.