  We examine the perturbative calculation of the inclusive semi-leptonic decay
rate $\Gamma$ for the $b$-quark, using mass-independent renormalization. To
finite order of perturbation theory the series for $\Gamma$ will depend on the
unphysical renormalization scale parameter $\mu$ and on the particular choice
of mass-independent renormalization scheme; these dependencies will only be
removed after summing the series to all orders. In this paper we show that all
explicit $\mu$-dependence of $\Gamma$, through powers of ln$(\mu)$, can be
summed by using the renormalization group equation. We then find that this
explicit $\mu$-dependence can be combined together with the implicit
$\mu$-dependence of $\Gamma$ (through powers of both the running coupling
$a(\mu)$ and the running $b$-quark mass $m(\mu)$) to yield a $\mu$-independent
perturbative expansion for $\Gamma$ in terms of $a(\mu)$ and $m(\mu)$ both
evaluated at a renormalization scheme independent mass scale $I\!\!M$ which is
fixed in terms of either the "$\overline{MS}$ mass" $\overline{m}_b$ of the $b$
quark or its pole mass $m_{pole}$. At finite order the resulting perturbative
expansion retains a degree of arbitrariness associated with the particular
choice of mass-independent renormalization scheme. We use the coefficients
$c_i$ and $g_i$ of the perturbative expansions of the renormalization group
functions $\beta(a)$ and $\gamma(a)$, associated with $a(\mu)$ and $m(\mu)$
respectively, to characterize the remaining renormalization scheme
arbitrariness of $\Gamma$. We further show that all terms in the expansion of
$\Gamma$ can be written in terms of the $c_i$ and $g_i$ coefficients and a set
of renormalization scheme independent parameters $\tau_i$.
