  The general relation between the standard expansion coefficients and the beta
function for the QCD coupling is exactly derived in a mathematically strict
way. It is accordingly found that an infinite number of logarithmic terms are
lost in the standard expansion with a finite order, and these lost terms can be
given in a closed form. Numerical calculations, by a new matching-invariant
coupling with the corresponding beta function to four-loop level, show that the
new expansion converges much faster.
