  In this work we illustrate a number of properties of the Born approximation
in the three-dimensional Calder\'on inverse conductivity problem by numerical
experiments. The results are based on an explicit representation formula for
the Born approximation recently introduced by the authors. We focus on the
particular case of radial conductivities in the ball $B_R \subset \mathbb{R}^3
$ of radius $R$, in which the linearization of the Calder\'on problem is
equivalent to a Hausdorff moment problem. We give numerical evidences that the
Born approximation is well defined for $L^{\infty}$ conductivities, and we
present a novel numerical algorithm to reconstruct a radial conductivity from
the Born approximation under a suitable smallness assumption. We also show that
the Born approximation has depth-dependent uniqueness and approximation
capabilities depending on the distance (depth) to the boundary $\partial B_R$.
We then investigate how increasing the radius $R$ affects the quality of the
Born approximation, and the existence of a scattering limit as $R\to \infty$.
Similar properties are also illustrated in the inverse boundary problem for the
Schr\"odinger operator $-\Delta +q$, and strong recovery of singularity results
are observed in this case.
