  We analyze the large-order behaviour in perturbation theory of classes of
diagrams with an arbitrary number of chains (i.e. photon lines, dressed by
vacuum polarization insertions). We derive explicit formulae for the leading
and subleading divergence as $n\to\infty$, and a complete result for the vacuum
polarization at the next-to-leading order in $1/N_f$. In general, diagrams with
more chains yield stronger divergence. We define an analogue of the familiar
diagrammatic $R$-operation, which extracts ultraviolet renormalon counterterms
as insertions of higher-dimension operators. We then use renormalization group
equations to sum the leading $(\ln n/N_f)^k$-corrections to all orders in
$1/N_f$ and find the asymptotic behaviour in $n$ up to a constant that must be
calculated explicitly order by order in $1/N_f$.
