  We study the ground state and the low-lying excitations of a trapped Bose gas
in an isotropic harmonic potential for very small ($\sim 3$) to very large
($\sim 10^7$) particle numbers. We use the correlated two-body basis functions
and the shape-dependent van der Waals interaction in our many-body
calculations. We present an exhaustive study of the effect of inter-atomic
correlations and the accuracy of the mean-field equations considering a wide
range of particle numbers. We calculate the ground state energy and the
one-body density for different values of the van der Waals parameter $C_{6}$.
We compare our results with those of the modified Gross-Pitaevskii results, the
correlated Hartree hypernetted-chain equations (which also utilize the two-body
correlated basis functions), as well as of the Diffusion Monte Carlo for hard
sphere interactions. We observe the effect of the attractive tail of the van
der Waals potential in the calculations of the one-body density over the truly
repulsive zero-range potential as used in the Gross-Pitaevskii equation and
discuss the finite-size effects. We also present the low-lying collective
excitations which are well described by a hydrodynamic model in the large
particle limit.
