  A very efficient large-order perturbation theory is formulated for the
nuclear motion of a linear triatomic molecule. To demonstrate the method, all
of the experimentally observed rotational energies, with values of $J$ almost
up to 100, for the ground and first excited vibrational states of CO$_2$ and
for the ground vibrational states of N$_2$O and of OCS are calculated. All
coupling between vibration and rotation is included. The perturbation
expansions reported here are rapidly convergent. The perturbation parameter is
$D^{-1/2}$, where $D$ is the dimensionality of space. Increasing $D$ is
qualitatively similar to increasing the angular momentum quantum number $J$.
Therefore, this approach is especially suited for states with high rotational
excitation. The computational cost of the method scales only as $JN_v^{5/3}$,
where $N_v$ is the size of the vibrational basis set.
