On the small--amplitude approximation to the differential equation $\ddot{x}+(1+\dot{x}^{2})x=0$

We obtain the radius of convergence of the small--amplitude approximation to the period of the nonlinear oscillator $\ddot{x}+(1+\dot{x}^{2})x=0$ with the initial conditions $x(0)=A$ and $\dot{x}(0)=0$ and show that the inverted perturbation series appears to converge smoothly from below.