  We study the field of an electric point charge that is slowly lowered into an
$n + 1$ dimensional Schwarzschild-Tangherlini black hole. We find that if $n >
3$, then countably infinite nonzero multipole moments manifest to observers
outside the event horizon as the charge falls in. This suggests the final state
of the black hole is not characterized by a Reissner-Nordstr\"om-Tangherlini
geometry. Instead, for odd $n$, the final state either possesses a degenerate
horizon, undergoes a discontinuous topological transformation during the infall
of the charge, or both. For even $n$, the final state is not guaranteed to be
asymptotically-flat.
