Is the formal energy of the mid-point rule convergent?

We obtain some formulae for calculation of the coefficients of four special types of terms in τ2k, k = 1, 2, … (1−1 corresponding to four type of (2k + 1)-vertex free unlabeled trees, k = 1, 2, …, respectively), for a fixed step size τ, in the tree-expansion of the formal energy of the mid-point rule. And, we give an estimate of the difference between the formal energy H and the standard Hamiltonian in some domain Ω under the assumptions 1. (i)H is smooth and bounded in Ω, and 2. (ii)the absolute values of the coefficients of the terms in τ2k are uniformly bounded by ησ2k for some constants η ≥ 1, σ > 0 and for any k ≥ 1.