A method is given which allows the presentation of the effective conductivity coefficient of a composite with a periodic structure in an analytical form as a continuous fraction expansion. Owing to rapid convergence, only a few levels of the fraction are needed to describe the effective conductivity coefficient with good accuracy. The obtained formula is valid for a wide range of parameters, including the asymptotic case when inclusions are very close to touching and the ratio of conductivities of both components tends to infinity. The detailed calculations have been carried out in the two-dimensional case for the square arrays of cylinders.