  Perturbative QCD corrections to hadronic $\tau$ decays and $e^+e^-$
annihilation into hadrons below charm are obtained from the Adler function,
which at present is known in the chiral limit to five-loop accuracy.
Extractions of the strong coupling, $\alpha_s$, from these processes suffer
from an ambiguity related to the treatment of unknown higher orders in the
perturbative series. In this work, we exploit the method of Pad\'e approximants
and its convergence theorems to extract information about higher-order
corrections to the Adler function in a systematic way. First, the method is
tested in the large-$\beta_0$ limit of QCD, where the perturbative series is
known to all orders. We devise strategies to accelerate the convergence of the
method employing renormalization scheme variations and the so-called D-log
Pad\'e approximants. The success of these strategies can be understood in terms
of the analytic structure of the series in the Borel plane. We then apply the
method to full QCD and obtain reliable model-independent predictions for the
higher-order coefficients of the Adler function. For the six-, seven-, and
eight-loop coefficients we find $c_{5,1} = 277\pm 51$, $c_{6,1}=3460\pm 690$,
and $c_{7,1}=(2.02\pm0.72)\times 10^4$, respectively, with errors to be
understood as lower and upper bounds. Our model-independent reconstruction of
the perturbative QCD corrections to the $\tau$ hadronic width strongly favours
the use of fixed-order perturbation theory (FOPT) for the renormalization-scale
setting.
