Find the sum of the series

Calculus Level 5

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

If the value of the sum above can be expressed in the form of \(\dfrac{ae}{b}\), where \(a\) and \(b\) are coprime positive integers, find \(a+b\).

\(\)
Clarification: \(e \approx 2.71828\) denotes the Euler's number.


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