The cuprate high temperature superconductors exhibit a pronounced trend in which the superconducting transition temperature, $T_{rm c}$, increases with the number of CuO$_2$ planes, $n$, in the crystal structure. We compare the magnetic excitation spectrum of Bi$_{2+x}$Sr$_{2-x}$CuO$_{6+delta}$ (Bi-2201) and Bi$_2$Sr$_2$Ca$_2$Cu$_3$O$_{10 + delta}$ (Bi-2223), with $n=1$ and $n=3$ respectively, using Cu $L_3$-edge resonant inelastic x-ray scattering (RIXS). Near the anti-nodal zone boundary we find the paramagnon energy in Bi-2223 is substantially higher than that in Bi-2201, indicating that multilayer cuprates host stronger effective magnetic exchange interactions, providing a possible explanation for the $T_{rm c}$ vs. $n$ scaling. In contrast, the nodal direction exhibits very strongly damped, almost non-dispersive excitations. We argue that this implies that the magnetism in the doped cuprates is partially itinerant in nature.
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