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

We calculate the low red-shift Taylor expansion for the luminosity distance for an observer at the center of a spherically symmetric matter inhomogeneity with a non vanishing 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 presence of a cosmological constant, without making any special assumption about the inhomogeneity profile.