  We show how the renormalons emerge from the renormalization group equation
with a priori no reference to any Feynman diagrams. The proof is rather given
by recasting the renormalization group equation as a resurgent equation studied
in the mathematical literature, which describes a function with an infinite
number of singularities in the positive axis of the Borel plane. Consistency
requires a one-to-one correspondence between the existence of such kind of
equation and the actual (generalized) Borel resummation of the renormalons
through a one-parameter transseries. Our finding suggests how non-perturbative
contributions can affect the running couplings. We also discuss these concepts
within the context of gauge theories, making use of the large number of flavor
expansion.
