Low-redshift formula for the luminosity distance in a LTB model with cosmological constant

We calculate the low-redshift Taylor expansion for the luminosity distance for an observer at the center of a spherically symmetric matter inhomogeneity with a nonvanishing cosmological constant. We then test the accuracy of the formulas comparing them to the numerical calculation for different cases for both the luminosity distance and the radial coordinate. The formulas can be used as a starting point to understand the general non-linear effects of a local inhomogeneity in the presence of a cosmological constant, without making any special assumption as regards the inhomogeneity profile.