Let ${S}_{n}=1\cdot (n-1)+2\cdot (n-2)+3\cdot (n-3)+\ldots +(n-1)\cdot 1,n\geq 4$. The sum $\sum _{n=4}^{\infty }\left(\frac{2{S}_{n}}{n!}-\frac{1}{(n-2)!}\right)$ is equal [JEE Main 2021, 1 Sep (Shift 2)]

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Let \({S}_{n}=1\cdot (n-1)+2\cdot (n-2)+3\cdot (n-3)+\ldots +(n-1)\cdot 1,n\geq 4\). The sum \(\sum _{n=4}^{\infty }\lef…

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