The usual Cauchy-Hadamard formula has a generalization to several variables.
The numbers $r_1,\ldots,r_n$ are called conjugate radii of convergence if the
series converges in the open polydisk $B(r_1,\ldots,r_n)$ and diverges in
$\{ z:|z_j|>r_j, 1\leq j\leq n\}$. Then we have the formula
$$\limsup_{|k|\to\infty}\left(|c_{k_1\ldots,k_n}|r_1^{k_1}\ldots r_n^{k_n}\right)^{|k|}=1,$$
where $|k|=k_1+\ldots+k_n.$

Reference: B. A. Fuks, Theory of analytic functions of several variables,
vol. I, Chap. I, sect 3, Theorem 3.7.