We propose an analytical solution of the problem of diffraction of a plane electromagnetic wave by a multilayer dielectric (including plasmon) sphere. The solution is obtained using the method of separation of variables. New efficient recurrence relationships are obtained for calculations of the fields in layers, as well as formulas for the fields in the near and far diffraction zones. The novelty of the proposed solution is connected with the way of representing its radial part in the form of normalized functions. It is shown that as the number of the lens layers, which approximate the smooth profile of dielectric permittivity, grows, the electric field at the focusing point increases and reaches the maximum value. This allows one to determine the minimum required number of layers in practical problems. Resonance properties of metal-dielectric nanoparticles are studied in the optical band.