We obtain the ground state magnetization of the Heisenberg and XXZ spin chains in a magnetic field $h$ as a series in $\sqrt {h_c - h}$ , where $h_c$ is the smallest field for which the ground state is fully polarized. All the coefficients of the series can be computed in closed form through a recurrence formula that involves only algebraic manipulations. For some values of the anisotropy parameter $\Delta$ the expansion is numerically observed to be convergent in the full range $0 \le h \le h_c$. To that end we express the free energy at mean magnetization per site $-1/2 \le \langle \sigma^z_i \rangle \le 1/2$ as a series in $1/2- \langle \sigma^z_i \rangle $ whose coefficients can be similarly recursively computed in closed form. This series converges for all $0 \le \langle \sigma^z_i \rangle \le 1/2$. The recurrence is nothing but the Bethe equations when their roots are written as a double series in their corresponding Bethe number and in $1/2- \langle \sigma^z_i \rangle $. It can also be used to derive the corrections in finite size, that correspond to the spectrum of a free compactified boson whose Luttinger parameter can be expanded as a similar series. The method presumably applies to a large class of models: it also successfully applies to a case where the Bethe roots lie on a curve in the complex plane.