Renormalizability
Higher order terms can be straightforwardly computed for the evolution operator but these terms display diagrams containing the following simpler ones

Oneloop contribution to the vacuum polarization function

Oneloop contribution to the electron selfenergy function

Oneloop contribution to the vertex function
that, being closed loops, imply the presence of diverging integrals having no mathematical meaning. To overcome this difficulty, a technique like renormalization has been devised, producing finite results in very close agreement with experiments. It is important to note that a criterion for theory being meaningful after renormalization is that the number of diverging diagrams is finite. In this case the theory is said renormalizable. The reason for this is that to get observables renormalized one needs a finite number of constants to maintain the predictive value of the theory untouched. This is exactly the case of quantum electrodynamics displaying just three diverging diagrams. This procedure gives observables in very close agreement with experiment as seen e.g. for electron gyromagnetic ratio.
Renormalizability has become an essential criterion for a quantum field theory to be considered as a viable one. All the theories describing fundamental interactions, except gravitation whose quantum counterpart is presently under very active research, are renormalizable theories.
Read more about this topic: Quantum Electrodynamics