  We study the first Steklov-Dirichlet eigenvalue on eccentric spherical shells
in $\mathbb{R}^{n+2}$ with $n\geq 1$, imposing the Steklov condition on the
outer boundary sphere, denoted by $\Gamma_S$, and the Dirichlet condition on
the inner boundary sphere. The first eigenfunction admits a Fourier--Gegenbauer
series expansion via the bispherical coordinates, where the
Dirichlet-to-Neumann operator on $\Gamma_S$ can be recursively expressed in
terms of the expansion coefficients arXiv:2309.09587. In this paper, we develop
a finite section approach for the Dirichlet-to-Neumann operator to approximate
the first Steklov--Dirichlet eigenvalue on eccentric spherical shells. We prove
the exponential convergence of this approach by using the variational
characterization of the first eigenvalue. Furthermore, based on the convergence
result, we propose a numerical computation scheme as an extension of the
two-dimensional result in [Hong et al., Ann. Mat. Pura Appl., 2022] to general
dimensions. We provide numerical examples of the first Steklov-Dirichlet
eigenvalue on eccentric spherical shells with various geometric configurations.
