# 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: