  We present a \emph{new} formulation of perturbation theory for quantum
systems, designated here as: `mean field perturbation theory'(MFPT), which is
free from power-series-expansion in any physical parameter, including the
coupling strength. Its application is thereby extended to deal with
interactions of \textit{arbitrary} strength and to compute system-properties
having non-analytic dependence on the coupling, thus overcoming the primary
limitations of the `standard formulation of perturbation theory' ( SFPT). MFPT
is defined by developing perturbation about a chosen input Hamiltonian, which
is exactly solvable but which acquires the non-linearity and the analytic
structure~(in the coupling-strength)~of the original interaction through a
self-consistent, feedback mechanism. We demonstrate Borel-summability of MFPT
for the case of the quartic- and sextic-anharmonic oscillators and the quartic
double-well oscillator (QDWO) by obtaining uniformly accurate results for the
ground state of the above systems for arbitrary physical values of the coupling
strength. The results obtained for the QDWO may be of particular significance
since `renormalon'-free, unambiguous results are achieved for its spectrum in
contrast to the well-known failure of SFPT in this case.
  \pacs{11.15.Bt,11.10.Jj,11.25.Db,12.38.Cy,03.65.Ge}
