Let `S_k,k=1, 2, …. 100` denote the sum of the infinite geometric series whose first term is `(k-1)/(K!)` and the common
ration is `1/k` then the value of `(100)^2/(100!)+underset(k=1)overset(100)Sigma`|(k^2-3k+1)S_k| is ____________`

Explanation

`S_(k)=(a)/(1-r)=((k-1)/(k!))/(1-(1)/(k))=(k)/(k!)=(1)/((k-1)!)`
Now, `sum_(k=2)^(100)|(k^(2)-3k+1)S_(k)|=sum_(k=2)^(100)|(k^(2)-3k+1)*(1)/((k-1)!)|`
`=sum_(k=2)^(100)|((k-1))/((k-2))=(k)/((k-1)!)|`
`=|(1)/(0!)-(2)/(1!)|+|(2)/(1!)-(3)/(2!)|+|(3)/(2!)-(4)/(3!)|+"........."+|(99)/(98!)-(100)/(99!)|`
`=((2)/(1!)-(1)/(0!))+((2)/(1!)-(3)/(2!))+((3)/(2!)-(4)/(3!))+"........."+((99)/(98!)-(100)/(99!))`
`=3-(100)/(99)=3-((100)^(2))/(100!)`
`:.((100)^(2))/(100!)+sum_(k=2)^(100)|(k^(2)-3k+1)S_(k)|=3`.