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The sum to infinity of the series : 1+...

The sum to infinity of the series `:`
`1+2 ( 1-(1)/( n )) + 3( 1 - ( 1)/( n ))^(2) +"...."` is `:`

A

`n^(2)`

B

`n(n+1)`

C

`n(1+(1)/(n))^(2)`

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
A
Let `S=1+2(1-(1)/(n))+3(1-(1)/(n))^(2)+"...."+oo`
`((1-(1)/(n))S=(1-(1)/(n))+2(1-(1)/(n))+"...."+oo)/(S(1-1+(1)/(n))=1(1-(1)/(n))+(1-(1)/(n))^(2)+"...."+oo)`
`implies (S)/(n)=(1)/(1-(1-(1)/(n))) " " [S_(oo)=(a)/(1-r) " by GP "]`
`implies S=(n)/((1)/(n))`
`implies S=n^(2)`.
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