Analysis of the ϵ-algorithm to accelerate the convergence

We study the ϵ-algorithm that was discussed in the papers of Guilpin et al. and Brezinski both published in 2004. Some researchers may motivate by their results to believe that is an efficient algorithm to generate many new series that will converge to the desired limit much faster than the original series. We will use the same example that appeared in Guilpin et al. to illustrate that some series derived by the ϵ-algorithm may not converge to the desired limit. Thus, we advise the researchers with care to look deeply into the existing literature before to use this sophisticated technique.