  The practical usefulness of Levin-type nonlinear sequence transformations as
numerical tools for the summation of divergent series or for the convergence
acceleration of slowly converging series, is nowadays beyond dispute. Weniger's
transformation, in particular, is able to accomplish spectacular results when
used to overcome resummation problems, often outperforming better known
resummation techniques, the most known being Pad\'e approximants. However, our
understanding of its theoretical features is still far from being satisfactory
and particularly bad as far as the decoding of factorially divergent series is
concerned.
  Stieltjes series represent a class of power series of fundamental interest in
mathematical physics. In the present paper, it is shown how the Stieltjes
series converging factor of any order is expressible as an inverse factorial
series, whose terms can be analytically retrieved through a simple recursive
algorithm. A few examples of applications of our algorithm are presented, in
order to show its effectiveness and implementation ease. We believe the results
presented here could constitute an important, preliminary step for the
development of a general convergence theory of Weniger's transformation on
Stieltjes series. A rather ambitious project, but worthy of being pursued in
the future.
