  From very high accuracy reflectivity spectra, we have derived the optical
conductivity and estimated the spectral weight up to various cut-off
frequencies in underdoped Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ (Bi-2212). We show
that, when evaluating the optical spectral weight over the full conduction band
(1 eV), the kinetic energy decreases in the superconducting state, unlike in
conventional BCS superconductors. As a consequence, the Ferrell-Glover-Tinkham
sum rule is not satisfied up to this energy scale. This stands as a very
unconventional behavior, contrasted with the overdoped Bi-2212 sample.
