  Persistence in coarsening 1D spin systems with a power law interaction
$r^{-1-\sigma}$ is considered. Numerical studies indicate that for sufficiently
large values of the interaction exponent $\sigma$ ($\sigma\geq 1/2$ in our
simulations), persistence decays as an algebraic function of the length scale
$L$, $P(L)\sim L^{-\theta}$. The Persistence exponent $\theta$ is found to be
independent on the force exponent $\sigma$ and close to its value for the
extremal ($\sigma \to \infty$) model, $\bar\theta=0.17507588...$. For smaller
values of the force exponent ($\sigma< 1/2$), finite size effects prevent the
system from reaching the asymptotic regime. Scaling arguments suggest that in
order to avoid significant boundary effects for small $\sigma$, the system size
should grow as ${[{\cal O}(1/\sigma)]}^{1/\sigma}$.
