The Born series for the scattering states reads
It can be derived by iterating the Lippmann–Schwinger equation
Note that the Green's operator  for a free particle can be retarded/advanced or standing wave operator for retarded  advanced  or standing wave scattering states .
The first iteration is obtained by replacing the full scattering solution  with free particle wave function  on the right hand side of the Lippmann-Schwinger equation and it gives the first Born approximation.
The second iteration substitutes the first Born approximation in the right hand side and the result is called the second Born approximation. In general the n-th Born approximation takes n-terms of the series into account. The second Born approximation is sometimes used, when the first Born approximation vanishes, but the higher terms are rarely used. The Born series can formally be summed as geometric series with the common ratio equal to the operator , giving the formal solution to Lippmann-Schwinger equation in the form