  We study the validity of the Neumann or Born series approach in solving the
Helmholtz equation and coefficient identification in related inverse scattering
problems. Precisely, we derive a sufficient and necessary condition under which
the series is strongly convergent. We also investigate the rate of convergence
of the series. The obtained condition is optimal and it can be much weaker than
the traditional requirement for the convergence of the series. Our approach
makes use of reduction space techniques proposed by Suzuki \cite{Suzuki-1976}.
Furthermore we propose an interpolation method that allows the use of the
Neumann series in all cases. Finally, we provide several numerical tests with
different medium functions and frequency values to validate our theoretical
results.
