The role of iteration, or the repeated application of an algorithm in accelerating the convergence of sequences, is investigated for certain well-known nonlinear transforms (like the ∊, u, θ and ϱ). It is found that the iterations of the Levin u transform improve upon the stability of the direct transform and are successful over a wide class of alternating, monotone and divergent series. An estimate of the numerical error in using iterations of the u2 (or θ 2) transform is given. A comparison of the estimated error with the actual error shows that the iterations of the u2 transform coupled with this error estimate provides a uniformly successful and, for most purposes, practical method of evaluating the limit of a sequence. A comparison of this method with other direct methods and their iterations is given over a wide class of alternating, monotone and divergent series.