It is shown that the application of linear and nonlinear algorithms improves the convergence of the series representing the doubly infinite free-space periodic Green's function. The numerical results indicate that the algorithms converge faster than the first-order acceleration. Convergence properties of the Green's function series are reported for the ‘on plane’ case in which the seriesjias the slowest convergence. The number of terms taken in the series and a relative error measure are given for various values of a convergence factor as the observation point is taken different locations within a unit cell.