Mathematics - Sequence & Series Question with Solution | TestHub

$\begin{array}{rl}& \mathrm{S}=1+\frac{1+3}{2!}+\frac{1+3+5}{3!}+\dots \\ & =\sum _{\mathrm{r}=1}^{\infty }\frac{{\mathrm{r}}^2}{\mathrm{r}!}\\ & =\sum _{\mathrm{r}=1}^{\infty }\frac{(\mathrm{r}-1+1)}{(\mathrm{r}-1)!}=\sum _{\mathrm{r}=2}^{\infty }\frac{1}{(\mathrm{r}-2)!}+\sum _{\mathrm{r}=1}^{\infty }\frac{1}{(\mathrm{r}-1)!}\\ & =2\mathrm{e}\end{array}$