$\pi$-Formulae from Dual Series of the Dougall Theorem

By means of the extended Gould-Hsu inverse series relations, we find that the dual relation of Dougall's summation theorem for the well--poised $_7F_6$-series can be utilized to construct numerous interesting Ramanujan--like infinite series expressions for $\pi^{\pm1}$ and $\pi^{\pm2}$, including an elegant formula of $\pi^{-2}$ due to Guillera.