\[\large 1+\frac{2^{2}}{2!}+\frac{3^{2}}{3!}+\frac{4^{2}}{4!}+\ldots\]

If the series above equals to \( n\times e\) for \(\displaystyle e =\lim_{x\to0} \left(1+\frac1x\right)^x \), find the value of \(n\).

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