  The operator product expansion (OPE), truncated in dimension, is employed in
many contexts. An example is the extraction of the strong coupling, $\alpha_s$,
from hadronic $\tau$-decay data, using a variety of analysis methods based on
finite-energy sum rules. Here, we reconsider a long-used method, which
parametrizes non-perturbative contributions to the $I=1$ vector and axial
vacuum polarizations with the OPE, setting several higher-dimension
coefficients to zero in order to implement the method in practice. The
assumption that doing this has a negligible effect on the value of $\alpha_s$
is tantamount to the assumption that the low-dimension part of the OPE
converges rapidly with increasing dimension near the $\tau$ mass. Were this
assumption valid, it would certainly have to be valid at energies above the
$\tau$ mass as well. It follows that the method can be tested using data
obtained from $e^+e^-\to\mbox{hadrons}$, as they are not limited by the
kinematic constraints of $\tau$ decays. We carry out such an investigation
using a recent high-precision compilation for the $R$-ratio, arguing that it
provides insights into the validity of the strategy, even if it probes a
different, though related channel. We find that $e^+e^-$-based tests call into
question the implied assumption of rapid convergence of the low-dimension part
of the OPE around the $\tau$ mass, and thus underscore the need to restrict
finite-energy sum-rule analyses to observables which receive only contributions
from lower-order terms in the OPE.
