  Exciting optical effects such as polarization control, imaging, and
holography were demonstrated at the nanoscale using the complex and irregular
structures of nanoparticles with the multipole Mie-resonances in the optical
range. The optical response of such particles can be simulated either by full
wave numerical simulations or by the widely used analytical coupled multipole
method (CMM), however, an analytical solution in the framework of CMM can be
obtained only in a limited number of cases. In this paper, a modification of
the CMM in the framework of the Born series and its applicability for
simulation of light scattering by finite nanosphere structures, maintaining
both dipole and quadrupole resonances, are investigated. The Born approximation
simplifies an analytical consideration of various systems and helps shed light
on physical processes ongoing in that systems. Using Mie theory and Green's
functions approach, we analytically formulate the rigorous coupled
dipole-quadrupole equations and their solution in the different-order Born
approximations. We analyze in detail the resonant scattering by dielectric
nanosphere structures such as dimer and ring to obtain the convergence
conditions of the Born series and investigate how the physical characteristics
such as absorption in particles, type of multipole resonance, and geometry of
ensemble influence the convergence of Born series and its accuracy.
