  We obtain a convergent power series expansion for the first branch of the
dispersion relation for subwavelength plasmonic crystals consisting of
plasmonic rods with frequency-dependent dielectric permittivity embedded in a
host medium with unit permittivity. The expansion parameter is $\eta=kd=2\pi
d/\lambda$, where $k$ is the norm of a fixed wavevector, $d$ is the period of
the crystal and $\lambda$ is the wavelength, and the plasma frequency scales
inversely to $d$, making the dielectric permittivity in the rods large and
negative. The expressions for the series coefficients (a.k.a., dynamic
correctors) and the radius of convergence in $\eta$ are explicitly related to
the solutions of higher-order cell problems and the geometry of the rods.
Within the radius of convergence, we are able to compute the dispersion
relation and the fields and define dynamic effective properties in a
mathematically rigorous manner. Explicit error estimates show that a good
approximation to the true dispersion relation is obtained using only a few
terms of the expansion. The convergence proof requires the use of properties of
the Catalan numbers to show that the series coefficients are exponentially
bounded in the $H^1$ Sobolev norm.
