  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.
