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Evaluate lim(ntooo)(1)/(n)[sin^(2k).(pi)...

Evaluate `lim_(ntooo)(1)/(n)[sin^(2k).(pi)/(2n)+sin^(2k).(2pi)/(2n)+....+sin^(2k).(pi)/(2)]`

Text Solution

Verified by Experts

`r^(th)term t_(r )=(1)/(n)*sin^(2k)(rpi)/(2n)`, where `r` varies from `1` to `n`
Given limit `=lim_(ntooo)sum_(r=1)^(n)(1)/(n)sin^(2k)(rpi)/(2n)`
`=int_(0)^(1)sin^(2k)((pi)/(2)x)dx`
`=(2)/(pi)int_(0)^((pi)/(2))((sin^(2k)t)dt` (Setting `t=(pi)/(2)x`)
`=(2)/(pi)*((2k-1))/(2k)*((2k-3))/((2k-2))..........(3)/(4)*(1)/(2)*(pi)/(2)`
`=((2k-1)(2k-3)......3.1)/(2k(2k-2)........4.2)`
`=((2k-1)(2k-3)......3.1)/(2^(k)k(k-1)......1)`
`=((2k-1)(2k-3).....3.1)/(2^(k)(k!))`
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