Based on the auxiliary equation method, the nonlinear superposition formula for the solutions of the second kind of elliptic equation is proposed. It is also used to construct the infinite sequence of new exact solutions to the generalized Boussinesq equation with the aid of symbolic computation system Mathematica. The infinite sequences of exact solutions include the Jacobi elliptic function infinite sequence solutions, the solitary wave infinite sequence solutions and the triangular function infinite sequence solutions. And the method is of significance to seek infinite sequence exact solutions to other nonlinear evolution equations.