  We present a simple and intuitive approximation for solving perturbation
theory (PT) of small cosmic fluctuations. We consider only the spherically
symmetric or monopole contribution to the PT integrals, which yields the exact
result for tree-graphs (i.e. at leading order). We find that the non-linear
evolution in Lagrangian space is then given by a simple {\it local}
transformation over the initial conditions, although it is not local in Euler
space. This transformation is found to be described the spherical collapse (SC)
dynamics, as it is the exact solution in the shearless (and therefore local)
approximation in Lagrangian space. Taking advantage of this property, it is
straightforward to derive the one-point cumulants, $\xi_J$, for both the
unsmoothed and smoothed density fields to {\it arbitrary} order in the
perturbative regime. To leading order this reproduces, and provides with a
simple explanation for, the exact results obtained by Bernardeau (1992, 1994).
We then show that the SC model leads to accurate estimates for the next
corrective terms when compared to the results derived in the exact perturbation
theory making use of the loop calculations (Scoccimarro & Frieman 1996). The
agreement is within a few per cent for the hierarchical ratios $S_J=
\xi_J/\xi_2^{J-1}$. We compare our analytic results to N-body simulations,
which turn out to be in very good agreement up to scales where $\sigma \approx
1$. A similar treatment is presented to estimate higher order corrections in
the Zel'dovich approximation. These results represent a powerful and readily
usable tool to produce analytical predictions to describe the gravitational
clustering of large scale structure in the weakly non-linear regime.
