As far as I've managed to understand this method, idea is the following. If we have equation like $x=1-a x^2$, then we assume that $a x^2$ is small and get zero approximation for the root $x_0 = 1$. We can calculate next orders based on Adomain polynomials and get the result:
$$x \to 1 - a + 2 a^2 - 5 a^3 + 14 a^4 - 42 a^5 + ...$$ (In fact what happens behind scenes here is that same as standard approaches pertrubation theory in physics.) For square equations I see that these series converge to correct root if square equation has a root. My question is if this works multivariate equations, i.e. we have multivariate nonlinear equation with at least one root, do we have a guarantee that Adomian series will converge?