Cool summation

Calculus Level 4

Let \(S_{k}\) ,\(k=1,2,3,....,100\), denote the sum of infinite G.P. whose first term is \(\frac { k-1 }{ k! }\) and common ratio is \(\frac { 1 }{ k }\). Then the value of \(\frac { { 100 }^{ 2 } }{ 100! } +\sum _{ k=1 }^{ 100 }{ \left| \left( { k }^{ 2 }-3k+1 \right) S_{k} \right| }\) is:

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