  We present the ab-plane optical conductivity of four single crystals of
Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\delta}$ (Bi2212) with different carrier doping
levels from the strongly underdoped to the strongly overdoped range with
$T_c$=66, 88, 77, and 67 K respectively. We focus on the redistribution of the
low frequency optical spectral weight (SW) in the superconducting and normal
states. The temperature dependence of the low-frequency spectral weight in the
normal state is significantly stronger in the overdoped regime. In agreement
with other studies, the superconducting order is marked by an increase of the
low frequency SW for low doping, while the SW decreases for the highly
overdoped sample. The effect crosses through zero at a doping concentration
$\delta$=0.19 which is slightly to the right of the maximum of the
superconducting dome. This sign change is not reproduced by the BCS model
calculations, assuming the electron-momentum dispersion known from published
ARPES data. Recent Cluster Dynamical Mean Field Theory (CDMFT) calculations
based on the Hubbard and t-J models, agree in several relevant respects with
the experimental data.
