Convergence acceleration for calculating radiated fields by a vertical electric dipole in the presence of a large sphere

The solution of the radiation of a vertical dipole in the presence of a large sphere is represented by infinite summation. It is known that the series usually has a convergence problem when both the source and the observation point are on or close to the surface. Also, the series converges slowly when the wavelength in air is much smaller than the radius of the sphere. The continuity of the field expressions on the spherical surface at r=r' in the space is discussed and we give a convergence acceleration method for the infinite summation due to a vertical electric dipole radiating in the presence of a perfectly conducting sphere.