Spooky + scary = Summation

Algebra Level 5

If Sk,(k=1,2,3,100),S_{k}, (k=1,2,3 \cdots ,100), denotes the infinite geometric progression sum, whose first term is k1k!\dfrac{k - 1}{k!} and common ratio is 1k\dfrac{1}{k} , then find (100)2100!+k=2100(k23k+1)Sk \large \dfrac{(100)^{2}}{100!} + \displaystyle \sum_{k=2}^{100} | (k^{2} - 3 k + 1) \cdot S_{k} |

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