  An approach is suggested defining effective sums of divergent series in the
form of self-similar exponential approximants. The procedure of constructing
these approximants from divergent series with arbitrary noninteger powers is
developed. The basis of this construction is the self-similar approximation
theory. Control functions governing the convergence of exponentially
renormalized series are defined from stability and fixed-point conditions and
from additional asymptotic conditions when the latter are available. The
stability of the calculational procedure is checked by analyzing cascade
multipliers. A number of physical examples for different statistical systems
illustrate the generality and high accuracy of the approach.
