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In this paper, we consider the Lane-Emden equation of the first kind which arises in the study of stellar structures. We use multiple algorithms, based on Homotopy Analysis Method (HAM) to find the convergent series solutions to the singular, non-linear, initial value problem. It is found that the radius of convergence for the solutions is affected by three factors: the choice of initial value, the order and type of non-linearity, and the linear operator used. We then compare analytical results to the 20th order series solution, the Pade approximant to the series, and the approximate solution obtained via the Runge-Kutta-Fehlberg method (RKF45).

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