Acceleration of convergence of some infinite sequences {An) whose asymptotic expansions involve fractional powers of n via the d(m) transformation

In this paper, we discuss the application of the author's d(m) transformation to accelerate the convergence of infinite series Enct la, when the terms an have asymptotic expansions that can be expressed in the form [m ioo an (n!)slm exp E as n oo, s integer. i=o i=o We discuss the implementation of the d(m) transformation via the recursive Walgorithm of the author. We show how to apply this transformation and how to assess in a reliable way the accuracies of the approximations it produces, whether the series converge or they diverge. We classify the different cases that exhibit unique numerical stability issues in floating-point arithmetic. We show that the d(m) transformation can also be used efficiently to accelerate the convergence of infinite products un + vo, where vn Etoein t/Tn i/rn as n Do, t > m + 1 an integer. Finally, we give several numerical examples that attest the high efficiency of the cl(m) transformation for the different cases.