Perturbation theory, asymptotic series and the renormalisation group

The divergence of the perturbation series in quantum field theory is reviewed and an argument presented showing that, at least in the unrenormalised theory, it must be asymptotic in the Poincaré sense. Some basic concepts of asymptotic series are reviewed. By combining momentum analyticity with the constraints of the renormalisation group it is shown how in quantum chromodynamics the large order coefficients can be estimated and the effective expansion parameter for the series derived. The optimum number of meaningful terms in the series is estimated.