_ n (1+ 1+1 2 + .+1 n n^ 2 )^ n =......... - Vidyadip

MCQ

$\lim _{n \rightarrow \infty}\left(1+\frac{1+\frac{1}{2}+\ldots \ldots .+\frac{1}{n}}{n^{2}}\right)^{n}=.........$

A
$\frac{1}{2}$
B
$0$
C
$\frac{1}{ e }$
✓
$1$

✓

Answer

Correct option: D.

$1$

d
Given limit is of $1^{\infty}$ form

$\text { So, } l=\exp \left(\lim _{n \rightarrow \infty} \frac{1+\frac{1}{2}+\frac{1}{3}+\ldots \ldots .+\frac{1}{n}}{n}\right)$

Now,

$0 \leq 1+\frac{1}{2}+\frac{1}{3}+\ldots .+\frac{1}{n} \leq 1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\ldots+\frac{1}{\sqrt{n}}$

$\leq 2 \sqrt{n}-1$

So, $l=\exp (0)$ (from sandwich theorem)

$=1$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "MCQ" from 12. limits→12. limits — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

If a circle of unit radius is divided into two parts by an arc of another circle subtending an angle $60^o$ on the circumference of the first circle, then the radius of the arc is

If $\text{|x}+2|\leq9,$ then:

A rhombus is inscribed in the region common to the two circles $x^2 + y^2 - 4x - 12 = 0$ and $x^2 + y^2 + 4x - 12 = 0$ with two of its vertices on the line joining the centres of the circles. The area of the rhombous is :

If in a frequency distribution, the mean and median are $21$ and $22$ respectively, then its mode is approximately

The quadrilateral formed by the vertices $(-1,1),\, (0,-3),\, (5,2)$ and $(4,6)$ will be

Any circle through the point of intersection of the lines $x + \sqrt 3 y = 1$ and $\sqrt 3 x - y = 2 $ if intersects these lines at points $P$ and $Q$, then the angle subtended by the arc $PQ$ at its centre is ............ $^o$

Let $\mathrm{m}$ and $\mathrm{n}$ be the coefficients of seventh and thirteenth terms respectively in the expansion of $\left(\frac{1}{3} \mathrm{x}^{\frac{1}{3}}+\frac{1}{2 \mathrm{x}^{\frac{2}{3}}}\right)^{18}$. Then $\left(\frac{\mathrm{n}}{\mathrm{m}}\right)^{\frac{1}{3}}$ is :

Let $\mathrm{X}$ be a random variable with distribution.

$\mathrm{x}$	$-2$	$-1$	$3$	$4$	$6$
$\mathrm{P}(\mathrm{X}=\mathrm{x})$	$\frac{1}{5}$	$\mathrm{a}$	$\frac{1}{3}$	$\frac{1}{5}$	$\mathrm{~b}$

If the mean of $X$ is $2.3$ and variance of $X$ is $\sigma^{2}$, then $100 \sigma^{2}$ is equal to :

The product of three geometric means between $4$ and $\frac{1}{4}$ will be

If the sum of $n$ terms of an $A.P.$ is $nA + {n^2}B$, where $A,B$ are constants, then its common difference will be