Acceleration of Convergence of Some Infinite Sequences $\{{\mathbf{A}}_{\mathbf{n}}\}$ Whose Asymptotic Expansions Involve Fractional Powers of $\mathbf{n}$ via the ${\tilde{d}}^{(m)}$ transformation

Abstract

In this paper, we discuss the application of the author's ${\tilde{d}}^{(m)}$ transformation to accelerate the convergence of infinite series ${\sum }_{n=1}^{\infty }{a}_n$ when the terms $a_n$ have asymptotic expansions that can be expressed in the form $$a_n\sim (n!{)}^{s/m}\exp \left[\sum _{i=0}^m{q}_i{n}^{i/m}\right]\sum _{i=0}^{\infty }{w}_i{n}^{\gamma -i/m}\text{as n→∞},s\text{integer.}$$ We discuss the implementation of the ${\tilde{d}}^{(m)}$ transformation via the recursive W-algorithm 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 ${\tilde{d}}^{(m)}$ transformation can also be used efficiently to accelerate the convergence of infinite products ${\prod }_{n=1}^{\infty }(1+v_n)$, where $v_n\sim {\sum }_{i=0}^{\infty }{e}_i{n}^{-t/m-i/m}$ as $n\to \infty$,\ $t\ge m+1$ an integer. Finally, we give several numerical examples that attest the high efficiency of the ${\tilde{d}}^{(m)}$ transformation for the different cases.