ABSTRACT
The exact calculation of the matrix elements is extraordinarily difficult beyond the second order in a perturbation expansion. Therefore, it is of crucial importance to examine the operator and vector series, with the purpose of gaining some insight into convergence of the pertinent physical matrix elements. A great advantage is that convergence of an operator expansion automatically guarantees the convergence of the corresponding series of vectors and matrix elements. The chapter examines in some detail a connection which exists between operator and vector expansions, thus establishing the quoted relation of the corresponding convergence radii. The importance of the convergence problem of the Neumann-Born expansions based upon the outlined concept of a comparative study of series of operators, vectors and matrix elements is in its application to the cases when the kernel K is not a compact operator.
