The Born series is the expansion of different scattering quantities in quantum scattering theory in the powers of the interaction potential  (more precisely in powers of  where  is the free particle Green's operator). It is closely related to Born approximation, which is the first order term of the Born series. The series can formally be understood as power series introducing the coupling constant by substitution . The speed of convergence and radius of convergence of the Born series are related to eigenvalues of the operator . In general the first few terms of the Born series are good approximation to the expanded quantity for "weak" interaction 
and large collision energy.