We analyze tau lepton dccay observables, namely the moments of the hadronic
spectral density in the finite energy interval (0, M-tau), within finite o
rder perturbation theory including alpha(s)(4) corrections. The start of th
e asymptotic growth of the perturbation theory series is found at this orde
r in a scheme-invariant manner. We establish the ultimate accuracy of finit
e order perturbation theory predictions and discuss the construction of opt
imal observables.