Summation of a Series!

Algebra Level 5

\[\large{ S = \sum_{k=1}^{2006} (-1)^k \dfrac{k^2 - 3}{(k+1)!}}\]

If \((S-1)\) can be represented as \(\dfrac{A}{B!}\) where \(A,B\) are positive integers each less than 10,000, then find the value of \(A+B\).

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