In this paper, we consider the multidimensional generalization of $S$-fraction, namely the multidimensional $S$-fraction with independent variables. This branched continued fraction with independent variables is an efficient tool for the approximation of multivariable functions, which are represented by formal multiple power series. We have investigated the convergence of multidimensional $S$-fraction with independent variables and have established the truncation error bounds for this branched continued fraction with independent variables in some domains of the space $\mathbb{C}^N$.

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