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We examine the QCD perturbation series at large orders, for different valuesof the 'large $\beta_0$ renormalization scale'. It is found that if we let thisscale grow exponentially with the order, the divergent series can be turnedinto 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 adimension four operator in the operator product expansion, this qualitativelyimproves the perturbative predictions. Furthermore, our results allow us toestablish formulations of the principle of minimal sensitivity and the fastestapparent convergence criterion that result in a convergent expansion.Comment: 14 pages, 5 figures, elaborated conclusion

physical review. d. particles, fields, gravitation, and cosmology/physical review. d, particles, fields, gravitation, and cosmology

QCD perturbation theory at large orders with large renormalization scales in the large<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math>limit

QCD perturbation theory at large orders with large renormalization scales in the largeβ0limit