We study periodically driven closed systems with a long-ranged Hamiltonian by considering a generalized Kitaev chain with pairing terms which decay with distance as a power law characterized by exponent $alpha$. Starting from an initial unentangled state, we show that all local quantities relax to well-defined steady state values in the thermodynamic limit and after $n gg 1$ drive cycles for any $alpha$ and driving frequency $omega$. We introduce a distance measure, $mathcal{D}_l(n)$, that characterizes the approach of the reduced density matrix of a subsystem of $l$ sites to its final steady state. We chart out the $n$ dependence of ${mathcal D}_l(n)$ and identify a critical value $alpha=alpha_c$ below which they generically decay to zero as $(omega/n)^{1/2}$. For $alpha > alpha_c$, in contrast, ${mathcal D}_l(n) sim (omega/n)^{3/2}[(omega/n)^{1/2}]$ for $omega to infty [0]$ with at least one intermediate dynamical transition. We also study the mutual information propagation to understand the nature of the entanglement spreading in space with increasing $n$ for such systems. We point out existence of qualitatively new features in the space-time dependence of mutual information for $omega < omega^{(1)}_c$, where $omega^{(1)}_c$ is the largest critical frequency for the dynamical transition for a given $alpha$. One such feature is the presence of {it multiple} light cone-like structures which persists even when $alpha$ is large. We also show that the nature of space-time dependence of the mutual information of long-ranged Hamiltonians with $alpha le 2$ differs qualitatively from their short-ranged counterparts with $alpha > 2$ for any drive frequency and relate this difference to the behavior of the Floquet group velocity of such driven system.
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