Acceleration of Convergence of Some Infinite Sequences $\boldsymbol{\{A_n\}}$ Whose Asymptotic Expansions Involve Fractional Powers of $\boldsymbol{n}$ 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 $\sum^\infty_{n=1}a_n$ when the terms $a_n$ have asymptotic expansions that can be expressed in the form $$ a_n\sim(n!)^{s/m}\exp\left[\sum^{m}_{i=0}q_in^{i/m}\right]\sum^\infty_{i=0}w_i n^{\gamma-i/m}\quad\text{as $n\to\infty$},\quad s\ \text{integer.}$$ 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 $\prod^\infty_{n=1}(1+v_n)$, where $v_n\sim \sum^\infty_{i=0}e_in^{-t/m-i/m}$ as $n\to\infty$,\ $t\geq m+1$ an integer. Finally, we give several numerical examples that attest the high efficiency of the $\tilde{d}^{(m)}$ transformation for the different cases.