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!)+ Sigma_(k=1)^(100)`|(k^2-3k+1)S_k| is ____________`

Answer

3

Explanation

`S_k=((k-1)/(k!))/(1-1/k)=1/((k-1)!)`
`underset(k=2)overset(100)Sigma |(k^2-3k+1)(1)/((k-1)!)|`
`underset(k=2)overset(100)Sigma |((k-1)^2-k)/((k-1)!)|`
`underset(k=2)overset(100)Sigma |(k-1)/((k-2)!)-k/((k-1)!)|`
`=|1/(0!)-2/(1!)|+|2/(1!)-3/(2!)|+|3/(2!)-(4)/(3!)|+....`
`=2/(1!)-1/(0!)+2/(1!)-3/(2!)+3/(2!)-4/(3!)+......+99/(98!)-100/(99!)`
`=3-100/(99!)`