Yet another question 4 - Corrected

Algebra Level 5

Let $a_k = (k^2 +1) k!$ and $b_n = \displaystyle \sum_{k=1}^{n} a_k$ .

$\dfrac{a_{100}}{b_{100}}$ can be represented as $\dfrac AB$ for positive coprime integers $A$ and $B$ .

Find $B-A$ .

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