The value of lim(n rarroo) sum(r=1)^(n)(...

The value of `lim_(n rarroo) sum_(r=1)^(n)(1)/(sin{((n+r)pi)/(4n)}).(pi)/(n) ` is equal to

A

`2ln(sqrt2-1)`

B

`4 ln (sqrt2-1)`

C

`4 ln (sqrt2+1)`

D

`ln sqrt2`

Verified by Experts

The correct Answer is:

C

`S=underset(nrarroo)(lim)sum_(r=1)^(n)(pi)/(n).(1)/(sin{(pi)/(4)+(pir)/(4n)})`
`" "=pi int_(0)^(1)(dx)/(sin((pi)/(4)x+(pi)/(4)))=4log_(e)(sqrt2-1)`

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