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In this paper, we present a generalization of the classical error bounds of Greengard-Rokhlin for the Fast Multipole Method (FMM) for Laplace potentials in three dimensions, extended to the case of local expansion (instead of point) targets. We also present a complementary, less sharp error bound proven via approximation theory whose applicability is not restricted to Laplace potentials. Our study is motivated by the GIGAQBX FMM, an algorithm for the fast, high-order accurate evaluation of layer potentials near and on the source layer. GIGAQBX is based on the FMM, but unlike a conventional FMM, which is designed to evaluate potentials at point-shaped targets, GIGAQBX evaluates local expansions of potentials at ball-shaped targets. Although the accuracy (or the acceleration error, i.e., error due to the approximation of the potential by the fast algorithm) of the conventional FMM is well understood, the acceleration error of FMM-based algorithms applied to the evaluation of local expansions has not been as well studied. The main contribution of this paper is a proof of a set of hypotheses first demonstrated numerically in the paper "A Fast Algorithm for Quadrature by Expansion in Three Dimensions," which pertain to the accuracy of FMM approximation of local expansions of Laplace potentials in three dimensions. These hypotheses are also essential to the three-dimensional error bound for GIGAQBX, which was previously stated conditionally on their truth and can now be stated unconditionally.