Let `S_k ,k=1,2, ,100 ,` denotes thesum of the infinite geometric series whose first term s `(k-1)/(k !)` and the common ratio is `1/k` , then the value of `(100^2)/(100 !)+sum_(k=2)^(100)(k^2-3k+1)S_k` is _______.

Options

A. 3
B. 6
C. 8
D. 9

Answer

A

Explanation

We have,
`S_(k)=((k-1)/(k!))/(1-(1)/(k))=(1)/((k-1)!)`
`:." "underset(k=2)overset(100)sum|(k^(2)-3k+1)S_(k)|`
`=underset(k=2)overset(100)sum|{:({(k-1)^(2)-k})/((k-1)!):}|`
`=underset(k=2)overset(100)sum|{:(k-1)/((k-2)!)-(k)/((k-1)!):}|`
`=underset(k=2)overset(100)sum|{:((k-2)+1)/((k-2)!)-((k-1)+1)/((k-1)!):}|`
`=underset(k=2)overset(100)sum|{:(1)/((k-3)!)-(1)/((k-2)!)+(1)/((k-2)!)-(1)/((k-2)!):}|`
`=underset(k=2)overset(100)sum{(1)/((k-3)!)-(1)/((k-2)!)}underset(k=2)overset(100)sum{(1)/((k-2)!)-(1)/((k-1)!)}`
`={(1-(1)/(1!))+((1)/(1!)-(1)/(2!))+((1)/(2!)-(1)/(3!))+ . . . +((1)/(97!)-(1)/(98!))}`
`+{(1-(1)/(1!))+((1)/(1!)-(1)/(2!))+((1)/(2!)-(1)/(3!))+ . . . +((1)/(98!)-(1)/(99!))}`
`=(1-(1)/(98!))+(1-(1)/(99!))=2(1)/(98!)-(1)/(99!)=2-(100)/(99!)`
Hence,
`((100)^(2))/(100!)+underset(k=1)overset(100)sum|{:(k^(2)-3k+1)S_(k):}|`
`(100)/(99!)+S_(1)+underset(k=2)overset(100)sum|{:(k^(2)-3k+1)S_(k):}|`
`=(100)/(99!)+1+2-(100)/(99!)=3`