Title: Acceleration of Convergence of Some Infinite Sequences $\boldsymbol{\{A_n\}}$ Whose Asymptotic Expansions Involve Fractional Powers of $\boldsymbol{n}$,
Abstract: In this paper, we deal with the acceleration of the convergence of infinite
series $\sum^\infty_{n=1}a_n$, when the terms $a_n$ are in general complex and
have asymptotic expansions that can be expressed in the form $$
a_n\sim[\Gamma(n)]^{s/m}\exp\left[Q(n)\right]\sum^\infty_{i=0}w_i
n^{\gamma-i/m}\quad\text{as $n\to\infty$},$$ where $\Gamma(z)$ is the gamma
function, $m\geq1$ is an arbitrary integer, $Q(n)=\sum^{m}_{i=0}q_in^{i/m}$ is
a polynomial of degree at most $m$ in $n^{1/m}$, $s$ is an arbitrary integer,
and $\gamma$ is an arbitrary complex number. This can be achieved effectively
by applying the $\tilde{d}^{(m)}$ transformation of the author to the sequence
$\{A_n\}$ of the partial sums $A_n=\sum^n_{k=1}a_k$, $n=1,2,\dots\ .$
We give a detailed review of the properties of such series and of the
$\tilde{d}^{(m)}$ transformation and the recursive W-algorithm that implements
it. We illustrate with several numerical examples of varying nature the
remarkable performance of this transformation on both convergent and divergent
series. We also show that the $\tilde{d}^{(m)}$ transformation can be used
efficiently to accelerate the convergence of some infinite products of the form
$\prod^\infty_{n=1}(1+v_n)$, where $$v_n\sim
\sum^\infty_{i=0}e_in^{-t/m-i/m}\quad \text{as $n\to\infty$,\ \ $t\geq m+1$ an
integer,}$$ and illustrate this with numerical examples. We put special
emphasis on the issue of numerical stability, we show how to monitor stability,
or lack thereof, numerically, and discuss how it can be achieved/improved in
suitable ways.