  The nonlinear Klein-Gordon (NLKG) equation on a manifold $M$ in the
nonrelativistic limit, namely as the speed of light $c$ tends to infinity, is
considered. In particular, a higher-order normalized approximation of NLKG
(which corresponds to the NLS at order $r=1$) is constructed, and when $M$ is a
smooth compact manifold or $\mathbb{R}^d$ it is proved that the solution of the
approximating equation approximates the solution of the NLKG locally uniformly
in time. When $M=\mathbb{R}^d$, $d \geq 3$, it is proved that solutions of the
linearized order $r$ normalized equation approximate solutions of linear
Klein-Gordon equation up to times of order $\mathcal{O}(c^{2(r-1)})$ for any
$r>1$.
