The value of the limit _ n = 2 ^ , ( 1 - 1 n^2 ) is - Vidyadip

MCQ

The value of the limit $\prod\limits_{n = 2}^\infty {\,\left( {1 - \frac{1}{{{n^2}}}} \right)} $ is

A
$1$
B
$\frac{1}{4}$
C
$\frac{1}{3}$
✓
$\frac{1}{2}$

✓

Answer

Correct option: D.

$\frac{1}{2}$

d
$\prod\limits_{n = 2}^\infty {\,\left( {\frac{{{n^2} - 1}}{{{n^2}}}} \right)} $=$\prod\limits_{n = 2}^\infty {\,\left( {\frac{{n - 1}}{n}} \right)} \,\,\prod\limits_{n = 2}^\infty {\,\left( {\frac{{n + 1}}{n}} \right)} $
=$\frac{{1\,\cdot\,2\,\cdot\,3........(n - 1)}}{{2\,\cdot\,3\,\cdot\,4......(n - 1)\,\cdot\,n}}$ =$\frac{1}{n}\,\cdot\,\frac{{n + 1}}{2}$ =$\mathop {Lim}\limits_{n \to \infty } \,\,\frac{{1 + \frac{1}{n}}}{2}$ =$\frac{1}{2}$

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

More "M.C.Q (1 Marks)" from STD 11 - 12. limits→STD 11 - 12. limits — all question types→Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

If the ratio of the lengths of tangents drawn from the point $(f,g)$ to the given circle ${x^2} + {y^2} = 6$ and ${x^2} + {y^2} + 3x + 3y = 0$ be $2 : 1$, then

$\frac{{{C_0}}}{1} + \frac{{{C_2}}}{3} + \frac{{{C_4}}}{5} + \frac{{{C_6}}}{7} + ....$=

A man wants to cut three lengths from a single piece of board of length $91 \ cm.$ The second length is to be $3 \ cm$ longer than the shortest and third length is to be twice as long as the shortest. What are the possible lengths for the shortest board if the third piece is to be at least $5 \ cm$ longer than the second?

The maximum distance from the origin of coordinates to the point $z$ satisfying the equation $\left| {z + \frac{1}{z}} \right| = a$is

The number of ways of selecting two numbers $a$ and $b$, $a \in\{2,4,6, \ldots ., 100\} \quad$ and $\quad b \in\{1,3,5, \ldots ., 99\}$ such that $2$ is the remainder when $a+b$ is divided by $23$ is$.......$

A bag contains $4$ white, $5$ red and $6$ black balls. If two balls are drawn at random, then the probability that one of them is white is

Let $A, B, C$ are three sets such that $n(A \cap B) = n(B \cap C) = n(C \cap A) = n(A \cap B \cap C) = 2$, then $n((A × B) \cap (B × C)) $ is equal to -

Equation of a line passing through the point of intersection of lines $2x - 3y + 4 = 0,$ $3x + 4y - 5 = 0$ and perpendicular to $6x - 7y + 3 = 0,$ then its equation is

Let $P\left(x_1, y_1\right)$ and $Q\left(x_2, y_2\right), y_1<0, y_2<0$, be the end points of the latus rectum of the ellipse $x^2+4 y^2=4$. The equations of parabolas with latus rectum $P Q$ are

$(A)$ $x^2+2 \sqrt{3} y=3+\sqrt{3}$

$(B)$ $x^2-2 \sqrt{3} y=3+\sqrt{3}$

$(C)$ $x^2+2 \sqrt{3} y=3-\sqrt{3}$

$(D)$ $x^2-2 \sqrt{3} y=3-\sqrt{3}$

If $\alpha,-\frac{\pi}{2}<\alpha<\frac{\pi}{2}$ is the solution of $4 \cos \theta+5 \sin \theta=1$, then the value of $\tan \alpha$ is