Question

$x_{n=}$$\sqrt{ \frac{1}{1^{2}*3 }- \frac{1}{1* 3^{2} } } + \sqrt{ \frac{1}{3^{2}*5 }- \frac{1}{3* 5^{2} } }+...+\sqrt{ \frac{1}{(2n-1)^{2}*(2n+1) }- \frac{1}{(2n-1)* (2n+1)^{2} } }$

Answer 1

$\frac{1}{(2n-1) ^{2} (2n+1)}- \frac{1}{(2n-1)(2n+1) ^{2} }$= \frac{2}{(2n-1) ^{2} *(2n+1)^{2}} [/tex] \fr$\lim_{n \to \infty} x_n = \lim_{n \to \infty} \frac{2}{1*3} + \frac{2}{3*5} +.........+ \frac{2}{(2n-1)(2n+1)} =$$= \lim_{n \to \infty}1- \frac{1}{3} + \frac{1}{3} - \frac{1}{5} +..........+ \frac{1}{2n-1} - \frac{1}{2n+1} = \lim_{n \to \infty} \frac{2n}{2n+1} =1$