  We examine the QCD perturbation series at large orders, for different values
of the 'large $\beta_0$ renormalization scale'. It is found that if we let this
scale grow exponentially with the order, the divergent series can be turned
into an expansion that converges to the Borel integral, with a certain cut off.
In the case of the first IR renormalon at $2/\beta_0$, corresponding to a
dimension four operator in the operator product expansion, this qualitatively
improves the perturbative predictions. Furthermore, our results allow us to
establish formulations of the principle of minimal sensitivity and the fastest
apparent convergence criterion that result in a convergent expansion.
