  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.
