  We investigate the convergence of relativistic hydrodynamics in charged
fluids, within the framework of holography. On the one hand, we consider the
analyticity properties of the dispersion relations of the hydrodynamic modes on
the complex frequency and momentum plane and on the other hand, we perform a
perturbative expansion of the dispersion relations in small momenta to a very
high order. We see that the locations of the branch points extracted using the
first approach are in good quantitative agreement with the radius of
convergence extracted perturbatively. We see that for different values of the
charge, different types of pole collisions set the radius of convergence. The
latter turns out to be finite in the neutral case for all hydrodynamic modes,
while it goes to zero at extremality for the shear and sound modes.
Furthermore, we also establish the phenomenon of pole-skipping for the
Reissner-Nordstrom black hole, and we find that the value of the momentum for
which this phenomenon occurs need not be within the radius of convergence of
hydrodynamics.
