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- W4297795287 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 $$ a_nsim(n!)^{s/m}expleft[sum^{m}_{i=0}q_in^{i/m}right]sum^infty_{i=0}w_i n^{gamma-i/m}quadtext{as $ntoinfty$},quad s text{integer.}$$ 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 $v_nsim sum^infty_{i=0}e_in^{-t/m-i/m}$ as $ntoinfty$, $tgeq m+1$ an integer. Finally, we give several numerical examples that attest the high efficiency of the $tilde{d}^{(m)}$ transformation for the different cases." @default.
- W4297795287 created "2022-10-01" @default.
- W4297795287 creator A5067073885 @default.
- W4297795287 date "2017-03-19" @default.
- W4297795287 modified "2023-10-14" @default.
- W4297795287 title "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" @default.
- W4297795287 doi "https://doi.org/10.48550/arxiv.1703.06495" @default.
- W4297795287 hasPublicationYear "2017" @default.
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