Limit _ n , , , , , 1 n , , , [ ,1 , , + , , n n , , + , ,1 , , ,… - Vidyadip

MCQ

$\mathop {Limit}\limits_{n\,\, \to \,\,\infty } \, \frac{1}{n}\,\,\,\left[ {\,1\,\, + \,\,\sqrt {\frac{n}{{n\,\, + \,\,1}}} \,\,\, + \,\,\,\sqrt {\frac{n}{{n\,\, + \,\,2}}} \,\,\, + \,\,\,\sqrt {\frac{n}{{n\,\, + \,\,3}}} \,\,\, + \,\,\,.......\,\,\, + \,\,\,\sqrt {\frac{n}{{n\,\, + \,\,3\,\,(n\,\, - \,\,1)}}} \,} \right]$ has the value equal to

A
$2 \,\sqrt 2$
B
$2\, \sqrt 2 - 1$
✓
$2$
D
$4$

✓

Answer

Correct option: C.

$2$

c
$T_r = \frac{1}{n}\,\,\sqrt {\frac{n}{{n\,\, + \,\,r}}}$

$\Rightarrow S =$ $\frac{1}{n}$$\sum\limits_{r\,\, = \,\,0}^{3\,n\,\, - \,\,3} {\,\,\sqrt {\frac{n}{{n\,\, + \,\,r}}} } $$= \int\limits_0^3 {\,\,\frac{1}{{\sqrt {1\,\, + \,\,x} }}} dx$

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