lim-n-r-1-n-1-1-n-n-r-n-r-

Question Number 17205 by Arnab Maiti last updated on 02/Jul/17

\lim_{n \to \infty} \underset{r = 1}{\sum^{n - 1}} \frac{1}{n} \sqrt{\frac{n + r}{n - r}}

Answered by ajfour last updated on 02/Jul/17

\frac{r}{n} \to x, \Rightarrow dx \to \frac{1}{n}

L = \int_{0}^{1} \sqrt{\frac{1 + x}{1 - x}} dx

let x = \cos 2 θ \Rightarrow dx = - 2 \sin θ \cos θ d θ

for x = 0, θ = π / 4

and when x = 1, θ = 0

since \sqrt{\frac{1 + \cos 2 θ}{1 - \cos 2 θ}} =∣ \frac{\cos θ}{\sin θ} ∣= \frac{\cos θ}{\sin θ}

in [0, π / 4]

so, L = \int_{π / 4}^{0} \frac{\cos θ}{\sin θ} (- 2 \sin θ \cos θ) d θ

L = \int_{0}^{π / 4} (1 + \cos 2 θ) d θ

= (θ + \frac{\sin 2 θ}{2}) ∣_{0}^{π / 4}

finally, L = \frac{π}{4} + \frac{1}{2} .

Commented by Arnab Maiti last updated on 02/Jul/17

Thank you sir .