A simple scheme is developed to compute the Green's function of a periodic semiinfinite array in free space. It is based on the spectral representation of the fields radiated by an infinite linear array of dipoles. Results related to successive linear arrays are added in the space domain. This summation can be accelerated tremendously by using an elementary extrapolation technique. The resulting formulation converges everywhere in the plane containing the array, and the number of terms required to achieve a given precision increases slowly away from this plane.