_ x , ( n n^2 , + 1^2 + n n^2 + 2^2 + n n^2 + 3^2 + ...1 5n ) is … - Vidyadip

MCQ

$\mathop {\lim }\limits_{x \to \infty } \,\left( {\frac{n}{{{n^2}\, + {1^2}}} + \frac{n}{{{n^2} + {2^2}}} + \frac{n}{{{n^2} + {3^2}}} + ...\frac{1}{{5n}}} \right)$ is equal to

A
$\frac{\pi }{4}$
B
$tan^{-1}\,\,(3)$
C
$\frac{\pi }{2}$
✓
$tan^{-1}\,\,(2)$

✓

Answer

Correct option: D.

$tan^{-1}\,\,(2)$

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

$\mathop {\lim }\limits_{x \to \infty } \sum\limits_{r = 1}^{2n} {\frac{1}{{\left( {1 + \frac{{{r^2}}}{{{n^2}}}} \right)}}} $ Using $D.I.$ as limit of sum, we get

$ = \int\limits_0^2 {\frac{{dx}}{{1 + {x^2}}} = {{\tan }^{ - 1}}2} $

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