  This paper studies the error introduced ($\displaystyle \varepsilon(x)$) by
the Stieltjes asymptotic approximation series $\displaystyle li_{*}(x) =
\frac{x}{\log(x)}\sum_{k=0}^{n-1}\frac{k!}{\log^{k}(x)} +
(\log(x)-n)\frac{xn!}{\log^{n+1}(x)}$ to the logarithmic integral function
$li(x)$ with $\displaystyle n = \lfloor \log(x) \rfloor$ for all $x\ge e$. For
this purpose, this paper uses some relations between term $\displaystyle
\frac{n!}{\log^{n}(x)}$ and Stirling's approximation formula. In particular,
this paper establishes two non-asymptotic lower and upper bounds for
$\varepsilon(e^n)$ with $n \ge 1$ and $1 \le m \le n$:
  (i) $\displaystyle \varepsilon(e^m) + \frac{\sqrt{2\pi}}{8}\sum_{k=m+1}^{n}
\frac{1}{k^{\frac{3}{2}}} \le \varepsilon(e^n) \le \varepsilon(e^{m}) +
\frac{\sqrt{2\pi}}{4}\sum_{k=m+1}^{n} \frac{k+2}{k^{\frac{5}{2}}}$
  (ii) $\displaystyle L_m - \frac{\sqrt{2\pi}}{4\sqrt{n+1}} \le \varepsilon_{n}
\le R_m - \frac{\sqrt{2\pi}}{2\sqrt{n}} - \frac{\sqrt{2\pi}}{3\sqrt{n^3}}$.
Here, $\displaystyle L_m = \varepsilon_{m} + \frac{\sqrt{2\pi}}{4\sqrt{m+1}}$
and $\displaystyle R_m = \varepsilon_{m} + \frac{\sqrt{2\pi}}{2\sqrt{m}} +
\frac{\sqrt{2\pi}}{3\sqrt{m^3}}$
  Moreover, this paper shows that if $\displaystyle |\varepsilon(x)| \le
\frac{\sqrt{2\pi}}{\sqrt{\log(x)}}$ then $li(e^n) \le li_{*}(e^n)$ for all $n
\ge 1$. Finally, this paper establishes non-asymptotic lower and upper bounds
for $\varepsilon(x)$ with $x \ge e$:
  $\displaystyle L_m - \frac{\sqrt{2\pi}}{4\sqrt{n+1}}
-\frac{\sqrt{2\pi}}{36\sqrt{n^3}} < \varepsilon(x) \le R_m -
\frac{\sqrt{2\pi}}{2\sqrt{n}} - \frac{\sqrt{2\pi}}{3\sqrt{n^3}}$
