  We study a simplified model of top production and decay, consisting in a
virtual vector boson $W^*$ decaying into a massive-massless $t$-$\bar{b}$
quark-antiquark pair. The top has a finite width and further decays into a
stable vector boson $W$ and a $b$ quark. We then consider the emission or the
virtual exchange of one gluon, with all possible light-quark loop insertions.
These are the dominant diagrams in the limit of an infinite number of light
flavours. We devise a procedure to compute this process fully, by analytic and
numerical methods, and for any infrared-safe final-state observables. We
examine the results at arbitrary orders in perturbation theory, and assess the
factorial growth associated with renormalons. We look for renormalon effects
leading to corrections of order $\Lambda_{\rm QCD}$, that we dub `linear'
renormalons, in the inclusive cross section (with and without selection cuts),
in the mass of the reconstructed-top system, and in the average energy of the
final-state $W$ boson, considering both the pole and the $\overline{\rm MS}$
scheme for the top mass. We find that the total cross section without cuts, if
expressed in terms of the $\overline{\rm MS}$ mass, does not exhibit linear
renormalons, but, as soon as selection cuts are introduced, jets-related linear
renormalons arise in any mass scheme. In addition, we show that the
reconstructed mass is affected by linear renormalons in any scheme and that the
average energy of the $W$ boson (that we consider as a simplified example of
leptonic observable), in any mass scheme, has a renormalon in the narrow-width
limit, that is however screened at large orders for finite top widths, provided
the top mass is in the $\overline{\rm MS}$ scheme.
