Further WZ-based methods for proving and generalizing Ramanujan's series

In 2002 and 2006, using a Wilf- Zeilberger-based method, Guillera introduced proofs for evaluations for what are considered as the simplest two series out of Ramanujan's 17 series for 1/pi. In this article, we show how the WZ method may be used in a fundamentally and nontrivially different way to prove these results, and to obtain identities for infinite families of Ramanujan-like series for 1/pi. We introduce a F-3(2)-recurrence that we had discovered experimentally, and we prove this recursion using the WZ method and apply it to obtain a series acceleration formula that we apply to formulate a new and simple proof for the Ramanujan series for 1/pi that has a convergence rate of 1/64, and we provide an infinite family of generalizations of this formula, and similarly for Ramanujan's series of convergence rate 1/4.