  For any integral operator $K$ in the Schatten--von Neumann classes of compact
operators and its approximated operator $K_N\sim(N\ge1)$ obtained by using for
example a quadrature or projection method, we show that the convergence of the
approximate $p$-modified Fredholm determinants
$\sideset{}{_{Np}}\det(I_N+zK_N)$ to the $p$-modified Fredholm determinants
$\sideset{}{_p}\det(I_\mathcal{H}+zK)$ is uniform for all $p\ge1$. As a result,
we give the rate of convergences when evaluating at an eigenvalue or at an
element of the resolvent set of $K$.
