  The leading term of the ground state energy/particle of a dilute gas of
bosons with mass $m$ in the thermodynamic limit is $2\pi \hbar^2 a \rho/m$ when
the density of the gas is $\rho$, the interaction potential is non-negative and
the scattering length $a$ is positive. In this paper, we generalize the upper
bound part of this result to any interaction potential with positive scattering
length, i.e, $a>0$ and the lower bound part to some interaction potentials with
shallow and/or narrow negative parts.
