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The reciprocal of the value of: ("lim")(...

The reciprocal of the value of: `("lim")_(ntooo)(1-1/(2^2))(1-1/(3^2))(1-1/(4^2))(1-1/(n^2))i s `

Text Solution

Verified by Experts

The correct Answer is:
`(2)`
We have
`L=underset(ntooo)limprod_(n=2)^(n)(n^(2)-1)/(n^(2))`
`=underset(ntooo)limprod_(n=2)^(n)(n-1)/(n).prod_(n=2)^(n)(n+1)/(n)`
`=underset(ntooo)lim((1)/(2).(2)/(3).(3)/(4)...(n-1)/(n))((3)/(2).(4)/(3).(5)/(4)...(n+1)/(n))`
`=underset(ntooo)lim(1)/(n).(n+1)/(2)=(1)/(2)`
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Knowledge Check

  • lim_(xto oo)((1)/(1-n^(2))+(2)/(1-n^(2))+ . . .+(n)/(1-n^(2))) is

    A
    0
    B
    `(-1)/2`
    C
    `(1)/(2)`
    D
    `-2`

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