The perturbative series used to extract alpha(s)(M-tau) from the tau hadron
ic width exhibits slow convergence. Asymptotic Pade-approximant and Pade su
mmation techniques provide an estimate of these unknown higher-order effect
s, leading to values for alpha(s)(M-tau) that are about 10% smaller than cu
rrent estimates. Such a reduction improves the agreement of alpha(s)(M-tau)
with the QCD evolution of the strong coupling constant from alpha(s)(M-z).