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Let S(k), where k = 1,2,....,100, denote...

Let `S_(k)`, where `k = 1,2`,....,100, denotes the sum of the infinite geometric series whose first term is `(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....

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The correct Answer is:
4
We have, `S_(k) = ((k-1)/(k!))/(1 - (1)/(k)) = (1)/((k -1)!)`
Now, `(k^(2) -3k +1)S_(k)= {(k -2) (k-1) -1} xx (1)/((k -1)!)`
`= (1)/((k -3)!) - (1)/((k -1)!)`
`rArr underset(k=1)overset(100)sum ! (k^(2) - 3k + 1) S_(k) |=1 + 1 +2 - ((1)/(99!) + (1)/(98!)) = 4 - (100^(2))/(100!)`
`rArr (100^(2))/(100!) + underset(k=1)overset(100)sum | (k^(2) -3k +1) S_(k)| = 4`
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