  Precise extractions of $\alpha_s$ from $\tau\to {\rm (hadrons)}+\nu_\tau$ and
from $e^+e^-\to {\rm (hadrons)}$ below the charm threshold rely on finite
energy sum rules (FESRs) where the experimental side is given by integrated
spectral function moments. Here we study the renormalons that appear in the
Borel transform of polynomial moments in the large-$\beta_0$ limit and in full
QCD. In large-$\beta_0$, we establish a direct connection between the
renormalons and the perturbative behaviour of moments often employed in the
literature. The leading IR singularity is particularly prominent and is behind
the fate of moments whose perturbative series are unstable, while those with
good perturbative behaviour benefit from partial cancellations of renormalon
singularities. The conclusions can be extended to QCD through a convenient
scheme transformation to the $C$-scheme together with the use of a modified
Borel transform which make the results particularly simple; the leading IR
singularity becomes a simple pole, as in large-$\beta_0$. Finally, for the
moments that display good perturbative behaviour, we discuss an optimized
truncation based on renormalisation scheme (or scale) variation. Our results
allow for a deeper understanding of the perturbative behaviour of integrated
spectral function moments and provide theoretical support for low-$Q^2$
$\alpha_s$ determinations.
