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

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