  Without Borel or Pad$\acute{e}$ techniques, we show that for a divergent
series with $n!$ large-order growth factor, the set of Hypergeometric series
$_{k+1}F_{k-1}$ represents suitable approximants for which there exist no free
parameters. The divergent $_{k+1}F_{k-1}$ series are then resummed via their
representation in terms of the Meijer-G function. The choice of $_{k+1}F_{k-1}$
accelerates the convergence even with only weak-coupling information as input.
For more acceleration of the convergence, we employ the strong-coupling and
large-order information. We obtained a new constraint that relates the
difference of numerator and denominator parameters in the Hypergeometric
approximant to one of the large-order parameters. To test the validity of that
constraint, we employed it to obtain the exact partition function of the
zero-dimensional $\phi^4$ scalar field theory. The algorithm is also applied
for the resummation of the ground state energies of $\phi_{0+1}^{4}$ and
$i\phi_{0+1}^{3}$ scalar field theories. We get accurate results for the whole
coupling space and the precision is improved systematically in using higher
orders. Precise results for the critical exponents of the $O(4)$-symmetric
field model in three dimensions have been obtained from resummation of the
recent six-loops order of the corresponding perturbation series. The recent
seven-loops order for the $\beta$-function of the $\phi^{4}_{3+1}$ field theory
has been resummed which shows non-existence of fixed points. The first
resummation result of the seven-loop series representing the fractal dimension
of the two-dimensional self-avoiding polymer is presented here where we get a
very accurate value of $d_f=1.3307$ compared to its exact value
($4/3\approx1.3333$).
