_n[(n)/(n^2+1^2)+(n)/(n^2+2^2)+(n)/(n^2+3^2)++15n] is equal to :

Question

$\displaystyle\lim_{n	o\infty}\left[\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2}+\frac{n}{n^2+3^2}+\cdots+\frac1{5n}ight]$ is equal to :

Accepted Answer

$$\mathop {\lim }\limits_{x \to \infty } \sum\limits_{r = 1}^{2n} {{n \over {{n^2} + {r^2}}}} $$

$$\mathop {\lim }\limits_{x \to \infty } \sum\limits_{r = 1}^{2n} {{1 \over {n\left( {1 + {{{r^2}} \over {{n^2}}}} \right)}} = \int\limits_0^2 {{{dx} \over {1 + {x^2}}}} }  = {\tan ^{ - 1}}2$$

Solutions · 1