  We show that the asymptotic $1/N$ expansion for the averages of linear
statistics of the GUE is convergent when the test function is an entire
function of order two and finite type. This allows to fully recover the mean
eigenvalue density function for finite $N$ from the coefficients of the
expansion thus providing a resummation procedure. As an intermediate result we
compute the bilateral Laplace transform of the GUE reproducing kernel in the
half-sum variable, generalizing a formula of Haagerup and Thorbj{\o}rnsen.
