  We present an analysis of the evolution of the Lyman-series forest into the
epoch of reionization using cosmological radiative transfer simulations in a
scenario where reionization ends late. We explore models with different
midpoints of reionization and gas temperatures. We find that once the
simulations have been calibrated to match the mean flux of the observed
Lyman-$\alpha$ forest at $4 < z < 6$, they also naturally reproduce the
distribution of effective optical depths of the Lyman-$\beta$ forest in this
redshift range. We note that the tail of the largest optical depths that is
most challenging to match corresponds to the long absorption trough of ULAS
J0148+0600, which we have previously shown to be rare in our simulations. We
consider the evolution of the Lyman-series forest out to higher redshifts, and
show that future observations of the Lyman-$\beta$ forest at $z>6$ will
discriminate between different reionization histories. The evolution of the
Lyman-$\alpha$ and Lyman-$\gamma$ forests are less promising as a tool for
pushing studies of reionization to higher redshifts due to the stronger
saturation and foreground contamination, respectively.
