  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.
