  The method of Fractional Borel Summation is suggested in conjunction with
self-similar factor approximants. The method used for extrapolating asymptotic
expansions at small variables to large variables, including the variables
tending to infinity, is described. The method is based on the combination of
optimized perturbation theory, self-similar approximation theory, and
Borel-type transformations. General Borel Fractional transformation of the
original series is employed. The transformed series is resummed in order to
adhere to the asymptotic power laws. The starting point is the formulation of
dynamics in the approximations space by employing the notion of
self-similarity. The flow in the approximation space is controlled, and
``deep'' control is incorporated into the definitions of the self-similar
approximants. The class of self-similar approximations, satisfying, by design,
the power law behavior, such as the use of self-similar factor approximants, is
chosen for the reasons of transparency, explicitness, and convenience. A
detailed comparison of different methods is performed on a rather large set of
examples, employing self-similar factor approximants, self-similar iterated
root approximants, as well as the approximation technique of self-similarly
modified Pade - Borel approximations.
