Using Fourier–Legendre expansions to derive series for $\frac{1}{\pi}$ and $\frac{1}{\pi^{2}}$

In this paper, we derive some series for $\frac{1}{\pi}$ and $\frac{1}{\pi^{2}}$ from the Fourier–Legendre expansions of odd powers of $\sqrt{1-x^{2}}$.