  Starting from the divergent character of the perturbative expansions in QCD
and using the technique of series acceleration by the conformal mappings of the
Borel plane, I define a novel, non-power perturbative expansion for the Adler
function, which simultaneously implements renormalization-group summation and
has a tamed large-order behaviour. The new expansion functions, which replace
the standard powers of the coupling, are singular at the origin of the coupling
plane and have divergent perturbative expansions, resembling the expanded
function itself. Confronting the new perturbative expansions with the standard
ones on specific models investigated recently in the literature, I show that
they approximate in an impressive way the exact Adler function and the spectral
function moments. Applied to the $\tau$ hadronic width, the contour-improved
and the renormalization-group summed non-power expansions in the
${\overline{\rm MS}}$ scheme lead to the prediction $\alpha_s(M_\tau^2)=
0.3192~^{+ 0.0167}_{-0.0126}$, which translates to $\alpha_s(M_Z^2)=
0.1184~^{+0.0020}_{-0.0016}$.
