  Based on the Sturm-Liouville eigenvalue problem, Banerjee \emph{et al.}
proposed a perturbative approach to analytically investigate the properties of
the ($2+1$)-dimensional superconductor with Born-Infeld electrodynamics [Phys.
Rev. D {\bf 87}, 104001 (2013)]. By introducing an iterative procedure, we will
further improve the analytical results and the consistency with the numerical
findings, and can easily extend the analytical study to the higher-dimensional
superconductor with Born-Infeld electrodynamics. We observe that the higher
Born-Infeld corrections make it harder for the condensation to form but do not
affect the critical phenomena of the system. Our analytical results can be used
to back up the numerical computations for the holographic superconductors with
various condensates in Born-Infeld electrodynamics.
