It is well-known since the work of Dyson, Bender and Wu, Lipatov,
Parisi and many other authors, that Feynman diagrams expansions yield
divergent series. In the case of lambda phi 4 quantum problems, I show
that by cutting off the large field configurations, one obtains
problems which are exponentially close to the usual ones, but the
corresponding series are converging. I show that this strategy can be
successfully applied to the anharmonic oscillator, and the
Landau-Ginzburg model in the hierarchical approximation. I discuss
the diagrammatic interpretation of the modified perturbative series.
I briefly discuss the possible extension of this method to gauge
theories.