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The reciprocal of the value of lim(nto...

The reciprocal of the value of
`lim_(ntooo) (1-(1)/(2^(2)))(1-(1)/(3^(2)))(1-(1)/(4^(2)))...(1-(1)/(n^(2)))`is__________.

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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