_ n [ 1 1 - n^2 + 2 1 - n^2 + 3 1 - n^2 + ........ + n 1 - n^2 ] … - Vidyadip

MCQ

$\mathop {\lim }\limits_{n \to \infty } \left[ {\frac{1}{{1 - {n^2}}} + \frac{2}{{1 - {n^2}}} + \frac{3}{{1 - {n^2}}} + ........ + \frac{n}{{1 - {n^2}}}} \right] =$

A
$0$
✓
$ - \frac{1}{2}$
C
$1/2$
D
None of these

✓

Answer

Correct option: B.

$ - \frac{1}{2}$

b
(b) $\mathop {\lim }\limits_{n \to \infty } \,\left[ {\frac{1}{{1 - {n^2}}} + \frac{2}{{1 - {n^2}}} + ..... + \frac{n}{{1 - {n^2}}}} \right]$

$ = \mathop {\lim }\limits_{n \to \infty } \,\frac{{\Sigma n}}{{1 - {n^2}}} = \frac{1}{2}\,\,\mathop {\lim }\limits_{n \to \infty } \,\,\,\frac{{{n^2} + n}}{{1 - {n^2}}} = - \frac{1}{2}$.

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

A circle of radius $2$ unit passes through the vertex and the focus of the parabola $y^{2}=2 x$ and touches the parabola $y=\left(x-\frac{1}{4}\right)^{2}+\alpha$, where $\alpha>0$.

Then $(4 \alpha-8)^{2}$ is equal to

The function $y = a(1 - \cos x)$ is maximum when $x = $

The locus of the point of intersection of lines $(x + y)t = a$ and $x - y = at$, where $t$ is the parameter, is

If $\alpha $ and $\beta $ are solutions of $sin^2\,x + a\, sin\, x + b = 0$ as well that of $cos^2\,x + c\, cos\, x + d = 0$ , then $sin\,(\alpha + \beta )$ is equal to

All the five digits numbers in which each successive digit exceeds its predecessor are arranged in the increasing order of their magnitude. The $97^{th}$ number in the list does not contain the digit:

Suppose that the points $(h, k), (1, 2)$ and $(-3, 4)$ lie on the line $L_1$. If a line $L_2$ passing through the points $(h, k)$ and $(4, 3)$ is perpendicular to $L_1$, then $\frac{k}{h}$ equals

Let $ABCD$ be a quadrilateral. If $E$ and $F$ are the mid points of the diagonals $AC$ and $BD$ respectively and $(\overrightarrow{ AB }-\overline{ BC })+(\overrightarrow{ AD }-\overrightarrow{ DC })= k \overline{ FE }$, then $k$ is equal to

For $\mathrm{x} \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, if $y(x)=\int \frac{\operatorname{cosec} x+\sin x}{\operatorname{cosec} x \sec x+\tan x \sin ^2 x} d x$ and $\lim _{x \rightarrow\left(\frac{\pi}{2}\right)^{-}} y(x)=0$ then $y\left(\frac{\pi}{4}\right)$ is equal to

If $f(x) = ax + b$ and $g(x) = cx + d$, then $f(g(x)) = g(f(x))$ is equivalent to

A pair has two children. If one of them is boy, then the probability that other is also a boy, is