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A numerical study of the multipole expansion for the multilevel fast multipole algorithm (MLFMA) is presented. In the numerical implementation of MLFMA, the error comes from three sources: the truncation of the addition theorem; the approximation of the integration; the aggregation and disaggregation process. These errors are due to the factorization of the Green's function which is the mathematical core of the algorithm. Among the three error sources, we focus on the truncation error and a new approach of selecting truncation numbers for the addition theorem is proposed. Using this approach, the error prediction and control can be improved for the small buffer sizes and high accuracy requirements.