  On the perturbatively non-renormalizable and non-perturbatively finite
examples (delta-function type potential in non-relativistic quantum mechanics
and the mathematical model of the propagator by Redmond and Uretsky in quantum
field theory) we illustrate that one can develop a perturbative approach for
non-renormalizable theory. The key idea is the introduction of finite number of
additional expansion parameters which allows us to eliminate all infinities
from the perturbative expressions. The generated perturbative series reproduce
the expansions of the exact analytical solutions.
