  The last years have seen significant developments in methods relating two-
and three-particle finite-volume energies to scattering observables. These
relations hold for both weakly and strongly interacting systems, and studying
their predictions in limiting cases can provide important cross checks as well
as giving useful insights into the general formulae. In these proceedings, we
present analytic results for finite-volume excited states, recovered by
expanding the general relations in powers of the interaction strength. We
highlight elegant patterns that emerge, especially for excited three-particle
energies, and discuss various applications of the results. The two-particle
results summarized here are described in more detail in the full manuscript,
and the three-particle results are detailed in a manuscript to appear.
