If | x | < 1, then _ n ( 1 + x ) ( 1 + x^2 ) ( 1 + x^4 )..... ( 1… - Vidyadip

MCQ

If $\left| x \right| < 1$, then $\mathop {\lim }\limits_{n \to \infty } \left\{ {\left( {1 + x} \right)\left( {1 + {x^2}} \right)\left( {1 + {x^4}} \right).....\left( {1 + {x^{2n}}} \right)} \right\}$ is equal to

A
$\frac{1}{{x - 1}}$
✓
$\frac{1}{{1 - x}}$
C
$1$
D
$x-1$

✓

Answer

Correct option: B.

$\frac{1}{{1 - x}}$

b
$\mathop {\lim }\limits_{n \to \infty } \left( {1 + x} \right)\left( {1 + {x^2}} \right)...\left( {1 + {x^{2n}}} \right)$

$ = \mathop {\lim }\limits_{n \to \infty } \frac{{\left( {1 - {x^{4n}}} \right)}}{{\left( {1 - x} \right)}} = \frac{{1 - 0}}{{1 - x}} = \frac{1}{{1 - x}}$

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