A description of the Thomas-Fermi ion with fast converging function series

This article concerns the description of the electron sea of an atomic ion with the Thomas-Fermi model. The normalized ion radius $X$, ionization potential $b$ and electronic binding energy $B$ of the Thomas-Fermi ion are functions of the ratio $N$ of electrons to protons in the ion. A scheme is given to calculate the Taylor series of $X(N)$, $b(N)$ and $B(N)$. With this scheme, the Taylor coefficients are calculated up to 5th order. The obtained 0th to 3rd order coefficients agree with the values presently available in the literature. To the authors knowledge, the 4th and 5th order coefficients are new results. It is then argued that a different series description of these functions, based on the Taylor series of $c(N) \equiv b^{-1/3}X^{-4/3}$, leads to a significant improvement in convergence.