  The properties of a generalized version of the Borel Transform in infrared
unstable theories with dynamical mass generation are studied. The
reconstruction of the nonperturbative structure is unambiguous in this version.
Various methods for extracting the singularity structure of the Borel Transform
for lattice formulations of such theories are explored, and illustrated
explicitly with the O(N) sigma model. The status of the first infrared
renormalon in QCD is discussed. The feasibility of a proposed technique for
analytically continuing from the left hand Borel plane (where nonperturbative
information is available via simulation of lattice field theory) to the
positive real axis is examined using the sigma model.
