  We discuss Pad\'e-improvement of known four-loop order results based upon an
asymptotic three-parameter error formula for Pad\'e-approximants. We derive an
explicit formula estimating the next-order coefficient $R_4$ from the previous
coefficients in a series $1+R_1 x + R_2x^2 + R_3x^3$. We show that such an
estimate is within 0.18% of the known five-loop order term in the O(1)
$\beta$-function, and within 10% of the known five-loop term in the O(1)
anomalous mass-dimension function $\gamma_m(g)$. We apply the same formula to
generate a [2$|$2] Pad\'e-summation of the QCD $\beta$-function and anomalous
mass dimension in order to demonstrate both the relative insensitivity of the
evolution of $\alpha_s(\mu)$ and the running quark masses to higher order
corrections, as well as a somewhat increased compatibility of the present
empirical range for $\alpha_s(m_\tau)$ with the range anticipated via evolution
from the present empirical range for $\alpha_s(M_z)$. For $3 \leq n_f \leq 6$
we demonstrate that positive zeros of any [2$|$2] Pad\'e-summation estimate of
the all-orders $\beta$-function which incorporates known two-, three-, and
four-loop contributions necessarily correspond to ultraviolet fixed points,
regardless of the unknown five-loop term. Pad\'e-improvement of higher-order
perturbative expressions is presented for the decay rates of the Higgs into two
gluons and into a $b \bar{b}$ pair, and is used to show the relative
insensitivity of these rates to higher order effects. However,
Pad\'e-improvement of the purely-perturbative component of scalar/pseudoscalar
current correlation functions is indicative of large theoretical uncertainties
in QCD sum rules for these channels, particularly if the continuum-threshold
parameter $s_0$ is near 1 GeV$^2$.
