  We prove that for divergent series solutions of nonlinear (or linear)
differential systems near a generic irregular singularity, the common
prescription of summation to the least term is, if properly interpreted,
meaningful and correct, and we extend this method to transseries solutions. In
every direction in the complex plane at the singularity (Stokes directions {\em
not} excepted) there exists a nonempty set of solutions whose difference from
the ``optimally'' (i.e., near the least term) truncated asymptotic series is of
the same (exponentially small) order of magnitude as the least term of the
series. There is a family of generalized Borel summation formulas $\mathcal{B}$
which commute with the usual algebraic and analytic operations (addition,
multiplication, differentiation, etc). We show that there is exactly one of
them, $\mathcal{B}_0$, such that for any formal series solution $\tilde{f}$,
$\mathcal{B}_0(\tilde{f})$ differs from the optimal truncation of $\tilde{f}$
by at most the order of the least term of $\tilde{f}$. We show in addition that
the Berry (1989) smoothing phenomenon is universal within this class of
differential systems. Whenever the terms ``beyond all orders'' {\em change} in
crossing a Stokes line, these terms vary smoothly on the Berry scale
$\arg(x)\sim |x|^{-1/2}$ and the transition is always given by the error
function; under the same conditions we show that Dingle's rule of signs for
Stokes transitions holds.
