An algebra problem by Manish Dash

Algebra Level 5

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


Please post the solution.

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