  Mellin-Barnes (MB) integrals are well-known objects appearing in many
branches of mathematics and physics, ranging from hypergeometric functions
theory to quantum field theory, solid state physics, asymptotic theory, etc.
Although MB integrals have been studied for more than one century, to our
knowledge there is no systematic computational technique of the multiple series
representations of $N$-fold MB integrals (for any given positive integer $N$).
We present here a breakthrough in this context, which is based on simple
geometrical considerations related to the gamma functions present in the
numerator of the MB integrand. The method rests on a study of $N$-dimensional
conic hulls constructed out of normal vectors of the singular (hyper)planes
associated with each of the gamma functions.
  Specific sets of these conic hulls are shown to be in one-to-one
correspondence with series representations of the MB integral. This provides a
way to express the series representations as linear combinations of multiple
series, the derivation of which does not depend on the convergence properties
of the latter. Our method can be applied to $N$-fold MB integrals with straight
as well as nonstraight contours, in the resonant and nonresonant cases and,
depending on the form of the MB integrand, it gives rise to convergent series
representations or diverging asymptotic ones.
  When convergent series are obtained the method also allows, in general, the
determination of a single ``master series'' for each linear combination, which
considerably simplifies convergence studies and numerical checks.
