  We systematically investigate different versions of variational perturbation
theory by forcing not only the first or second but also higher derivatives of
the approximant with respect to the variational parameter to vanish. The choice
of the highest derivative version turns out to be the most successful one for
approximating the ground-state energy of the anharmonic oscillator. It is
therefore used to determine the critical exponent alpha of the specific heat in
superfluid 4He in agreement with the value measured in recent space shuttle
experiments.
