First slide

Introduction to limits

Question

$If | x | < 1 then \underset{n \to \infty}{L t} (1 + x) (1 + x^{2}) (1 + x^{4}) (1 + x^{8}) ....... (1 + x^{2 n - 1}) =$

Moderate

A

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

$\frac{1}{1 + x}$
C

1
D

0

Solution

$\underset{n \to \infty}{L t} (1 + x) (1 + x^{2}) \cdot \cdot \cdot \cdot (1 + x^{2 n}) = \underset{n \to \infty}{L t} \frac{1}{1 - x} (1 - x) (1 + x) \cdot \cdot \cdot (1 + x^{2 n})$
$= \underset{n \to \infty}{L t} \frac{1}{1 - x} (1 - x^{4 n}) = \frac{1 - 0}{1 - x} = \frac{1}{1 - x}$ $(\sin c e |x| < 1)$

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