Assertion (A) : $1+\frac{3}{2!}+\frac{5}{4!}+\frac{7}{6!}+\dots \dots =\mathrm{e}$

Reason (R) : The coefficent of x² in $1+\frac{2+3\mathrm{x}}{1!}+\frac{{(2+3\mathrm{x})}^2}{2!}+\dots \dots \infty \mathrm{is} \frac{9{\mathrm{e}}^2}{2}$

Assention :

$\begin{array}{rcl} & & 1+\frac{3}{\overline{)\underline{2}}}+\frac{5}{\overline{)\underline{4}}}+\frac{7}{\overline{)\underline{6}}}+....\\ & =& \sum _0^{\infty } \frac{(2\mathrm{n}+1)}{\overline{)\underline{2\mathrm{n}}}}\\ & =& \sum _0^{\infty } \left(\frac{1}{\overline{)\underline{2\mathrm{n}-1}}}+\frac{1}{\overline{)\underline{2\mathrm{n}}}}\right)\\ & =& (0+1)+\left(\frac{1}{\overline{)\underline{1}}}+\frac{1}{\overline{)\underline{2}}}\right)+...\\ & =& \mathrm{e}\end{array}$

Reason :

$\begin{array}{rcl} & & 1+\frac{2+3\mathrm{x}}{\overline{)\underline{1}}}+\frac{{(2+3\mathrm{x})}^2}{\overline{)\underline{2}}}+....\infty \\ & =& {\mathrm{e}}^{2+3\mathrm{x}}={\mathrm{e}}^2.{\mathrm{e}}^{3\mathrm{x}}\\ & =& {\mathrm{e}}^2\left(1+3\mathrm{x}+\frac{{\left(3\mathrm{x}\right)}^2}{\overline{)\underline{2}}}+....\right)\end{array}$
$\begin{array}{rcl}{\mathrm{x}}^2 \mathrm{coef}& =& {\mathrm{e}}^2\left(\frac{3^2}{\overline{)\underline{2}}}\right)\\ & =& \frac{9{\mathrm{e}}^2}{2}\end{array}$