Download App

STD 11 - 12. limits — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 11 - 12. limits4 Marks
Question
$\lim _{n \rightarrow \infty}\left(1+\frac{1+\frac{1}{2}+\ldots \ldots .+\frac{1}{n}}{n^{2}}\right)^{n}=.........$
✓

Answer

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

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

The number of real roots of the equation, $\mathrm{e}^{4 \mathrm{x}}+\mathrm{e}^{3 \mathrm{x}}-4 \mathrm{e}^{2 \mathrm{x}}+\mathrm{e}^{\mathrm{x}}+1=0$ is 
→
Let $M$  and  $m$  be respectively the local maximum and the local minimum values of the function, $f(x) = \,2{x^3} - 9{x^2} + 12x + 5$  in the interval $[0, 3].$  Then $M-m$  is equal to
→
Let $\mathrm{a}>0$ be a root of the equation $2 \mathrm{x}^2+\mathrm{x}-2=0$.

If $\lim _{x \rightarrow \frac{1}{a}} \frac{16\left(1-\cos \left(2+x-2 x^2\right)\right)}{\left(1-a x^2\right)}=\alpha+\beta \sqrt{17}$, where $\alpha, \beta \in Z$ then $\alpha+\beta$ is equal to....................

→
The number of solutions of the equation $2 \theta-\cos ^{2} \theta+\sqrt{2}=0$ is $R$ is equal to
→
Consider a square matrix of order $5$ such that ${a_{ij}} = 0\,\,\forall \,\,i + j\, = n + 1,\,a_{ij}\, \in \left\{ {0,1} \right\}\,\,\forall \,\,i,j$ . In each row as well as in each column there in only one non-zero element. Then number of such matrices is
→
In a triangle $ABC$, coordianates of $A$ are $(1, 2)$ and the equations of the medians through $B$ and $C$ are $x + y = 5$ and $x = 4$ respectively. Then area of $\Delta ABC$ (in sq. units) is
→
Let $\alpha=1^2+4^2+8^2+13^2+19^2+26^2+\ldots \ldots$ upto 10 terms and $\beta=\sum_{n=1}^{10} n^4$. If $4 \alpha-\beta=55 k+40$, then $\mathrm{k}$ is equal to____________.
→
Let $P_{1}, P_{2}, \ldots \ldots, P_{15}$ be $15$ points on a circle. The number of distinct triangles formed by points $P_{i}, P_{j}, P_{k}$ such that $i+j+k \neq 15$, is :
→
The number of integers, between $100$ and $1000$ having the sum of their digits equals to $14$ , is $............$
→
The remainder on dividing $5^{99}$ by $11$ is
→