Let $S_{n} = \frac{1}{3!} + \frac{5}{4!} + \frac{11}{5!} + \frac{19}{6!} + \ldots. +$ upto $n$ terms

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Let $S_{n} = \frac{1}{3!} + \frac{5}{4!} + \frac{11}{5!} + \frac{19}{6!} + \ldots. +$ upto $n$ terms and $S = \lim_{n \rightarrow \infty}\mspace{2mu} S_{n}$, then

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Sol. [Key Concept: Sum to $n$ terms in closed form.]$$S_{n} = \sum_{r = 1}^{n}\mspace{2mu}\frac{r^{2} + r - 1}{(r + 2)!} = \sum_{r = 1}^{n}\mspace{2mu}\frac{r^{2} + 2r - (r + 1)}{(r + 2)!} = \sum_{r = 1}^{n}\mspace{2mu}\left( \frac{r}{(r + 1)!} - \frac{r + 1}{(r + 2)!} ight)$$$$\begin{matrix} & \ = \frac{1}{2!} - \frac{n + 1}{(n + 2)!} \ \Rightarrow \ S & \ = \frac{1}{2} \end{matrix}$$✘ (a) $S = \frac{12}{
eq}\frac{16}{}$✘ (b) not the closed form✔ (c) $S_{n} = \frac{1}{2!} - \frac{n + 1}{(n + 2)!}$✔ (d) $S = \frac{12}{}$

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