Convergence of the hydrodynamic gradient expansion in relativistic kinetic theory

We rigorously prove that, in any relativistic kinetic theory whose nonhydrodynamic sector has a finite gap, the Taylor series of all hydrodynamic dispersion relations has a finite radius of convergence. Furthermore, we prove that, for shear waves, such radius of convergence cannot be smaller than $1/2$ times the gap size. Finally, we prove that the nonhydrodynamic sector is gapped whenever the total scattering cross section (expressed as a function of the energy) is bounded below by a positive nonzero constant. These results, combined with well-established covariant stability criteria, allow us to derive a rigorous upper bound on the shear viscosity of relativistic dilute gases.