  We prove that Pade approximants yield increasingly accurate predictions of
higher-order coefficients in QCD perturbation series whose high-order behaviour
is governed by a renormalon. We also prove that this convergence is accelerated
if the perturbative series is Borel transformed. We apply Pade approximants and
Borel transforms to the known perturbative coefficients for the Bjorken sum
rule. The Pade approximants reduce considerably the renormalization-scale
dependence of the perturbative correction to the Bjorken sum rule. We argue
that the known perturbative series is already dominated by an infra-red
renormalon, whose residue we extract and compare with QCD sum-rule estimates of
higher-twist effects. We use the experimental data on the Bjorken sum rule to
extract $\alpha_s(M_Z^2) = 0.116_{-0.006}^{+0.004}$, including theoretical
errors due to the finite order of available perturbative QCD calculations,
renormalization-scale dependence and higher-twist effects.
