Infinite Factorial Summation with Sum of Squares

Calculus Level 4

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

If the above series can be expressed as \(S\), then find the value of \(\big \lfloor 100S \rfloor\) .

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