$$\begin{gathered} \mathrm{P}=1-\frac{3}{1!}+\frac{9}{2!}-\frac{27}{3!}+\dots \dots \dots \\ \mathrm{Q}=1+\frac{4}{1!}+\frac{16}{2!}+\frac{64}{3!}+\dots \dots \dots \\ \mathrm{R}={\log }_{\mathrm{e}}^3+\frac{{\left({\log }_{\mathrm{e}}^3\right)}^2}{2!}+\frac{{\left({\log }_{\mathrm{e}}^3\right)}^3}{3!}+\dots \dots \dots \end{gathered}$$

The ascending order of P, Q, R

$\begin{array}{rcl}\mathrm{P}& =& 1-\frac{3}{\overline{)\underline{1}}}+\frac{3^2}{\overline{)\underline{2}}}-\frac{3^3}{\overline{)\underline{3}}}+....\\ & =& {\mathrm{e}}^{-3}\\ \mathrm{Q}& =& 1+\frac{4}{\overline{)\underline{1}}}+\frac{4^2}{\overline{)\underline{2}}}+\frac{4^3}{\overline{)\underline{3}}}+...\\ & =& {\mathrm{e}}^4\\ \mathrm{R}& =& {\log }_{\mathrm{e}}^3+\frac{{\left({\log }_{\mathrm{e}}^3\right)}^2}{\overline{)\underline{2}}}+\frac{{\left({\log }_{\mathrm{e}}^3\right)}^3}{\overline{)\underline{3}}}+....\\ & =& {\mathrm{e}}^{{\log }_{\mathrm{e}}^3}-1=3-1=2\end{array}$

Ascending order ${\mathrm{e}}^{-3}, 2, {\mathrm{e}}^4$

                   P, R, Q