Applications of Tannery's theorem to Bailey's ψ66 summation

Ramanujan's ψ11 summation is obtained from Bailey's ψ66 summation by taking a limit and then specializing one of the variables. The process is shown to yield a three-term transformation for the general ψ22 series as an intermediate stage. Tannery's theorem is applied carefully, leading to the identification of two counterintuitive instances of series for which interchanging the limit and the summation yields incorrect results. Analogous limiting processes involving ordinary hypergeometric series are considered.