  We study the asymptotic behaviour of the perturbative series in the heavy
quark effective theory (HQET) using the $1/N_f$ expansion. We find that this
theory suffers from an {\it ultraviolet} renormalon problem, corresponding to a
non-Borel-summable behaviour of perturbation series in large orders, and
leading to a principal nonperturbative ambiguity in its definition. This
ambiguity is related to an {\it infrared} renormalon in the pole mass and can
be understood as the necessity to include the residual mass term $\delta m$ in
the definition of HQET, which must be considered as ambiguous (and possibly
complex), and is required to cancel the ultraviolet renormalon singularity
generated by the perturbative expansion. The formal status of $\delta m$ is
thus identical to that of condensates in the conventional short-distance
expansion of correlation functions in QCD. The status of the pole mass of a
heavy quark, the operator product expansion for inclusive decays, and QCD sum
rules in the HQET are discussed in this context.
