  The usual strategy for deducing the $\pi\mbox{--}\pi^\ast$ electronic energy
(or optical bandgap) in a molecule with an "infinite" number of conjugated
double bonds consists in fitting a function with some adjustable parameters to
the relevant data for a set of homologous molecules with increasing number of
repeat units ($N$), and assuming that, after its parameters have been optimized
according to the least-squares criterion, the function can be extended
indefinitely, and its output will coincide with, or come close to, the correct
limit. Since more than ten homologues are seldom available, one might wonder
whether extrapolation to the infinite-mer upon such slender basis is an
instance of sound inductive reasoning or a mere leap of faith. The present
article argues that the shape of the fitting function is an equally important
criterion, and points out that the expressions proposed by Hirayama and by
Meier and coworkers are flat functions of $1/N$, and that such functions are
incongruent with the currently available evidence. Formulas derived by Davydov
and by W. Kuhn are shown to be special cases of a new equation that outperforms
both.
