  We consider the structure of the leading ultra-violet (UV) renormalon
singularity associated with the QCD vacuum polarization Adler D-function, in
the approximation that only planar Feynman diagrams are retained. This ``planar
approximation'' results in some simplifications, in particular three of the
four potential contributions from four-fermion operators are shown to be
absent. We are able to obtain a fully normalized result for the leading
$n\to\infty$ behaviour of the portion of perturbative coefficients proportional
to $N_f^{n-r} N^r$, for SU(N) QCD with $N_f$ quark flavours.
