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The authors focus on the efficient computation of the slowly convergent infinite series that lead to the off-diagonal elements of the vector potential multilayered periodic dyadic Green's function. Two different approaches based on Kummer's transformation are applied to the evaluation of these series. The well-known approach that makes use of the generalized pencil of functions (GPoF) and Ewald's method is the fastest approach, but it does not provide accurate results when the distance between the field point and any of the source points is close to zero. To avoid this problem, we present a novel approach based on the GPoF and the spectral Kummer-Poisson's method with higher-order asymptotic extraction. This latter approach is slightly slower than the former one, but it is accurate in the whole range of distances between the field point and the sources.