  We employ a high-order perturbative expansion to characterize the ground
state of the Mott phase of the one-dimensional Bose-Hubbard model. We compute
for different integer filling factors the energy per lattice site, the
two-point and density-density correlations, and expectation values of powers of
the on-site number operator determining the local atom number fluctuations
(variance, skewness, kurtosis). We compare these expansions to numerical
simulations of the infinite-size system to determine their range of
applicability. We also discuss a new sum rule for the density-density
correlations that can be used in both equilibrium and non-equilibrium systems.
