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 val

Question

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 val

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 _______.

Monjur Alahi

8 views

2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

`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`.