The solution of a quantum periodic potential in solid state physics is relevant because one can understand how electrons behave in a corresponding crystal. In this paper, the analytical solution of the energy formulation for a one-dimensional periodic potential that meets certain boundary conditions is developed in a didactic and detailed way. In turn, the group speed and effective mass are also calculated from the transcendental energy equation of a potential $V=V\left(x\right)$. From this, a comparison is made with periodic potentials with known analytical solutions, such as the Dirac delta, as well as rectangular and triangular potentials. Finally, some limits are proposed in which these periodic potentials of the form $V=V\left(x\right)$ approach the periodic Dirac delta potential of positive intensity. Therefore, the calculations described in this paper can be used as the basis for more-complex one-dimensional potentials and related simulations.