First slide

Introduction to limits

Question

$If | x | < 1 then {Lt}_{n \to \infty} (1 + x) (1 + x^{2}) (1 + x^{4}) (1 + x^{8}) \dots . . (1 + x^{2 n - 1}) =$

Moderate

A

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

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

1
D

$1 + x$

Solution

$\begin{matrix} \underset{n \to \infty}{L t} (1 + x) (1 + x^{2}) \cdot \cdot \cdot \cdot (1 + x^{2 n - 1}) = \underset{n \to \infty}{L t} \frac{1}{1 - x} (1 - x) (1 + x) \cdot \cdot \cdot \cdot (1 + x^{2 n - 2}) (1 + x^{2 n - 1}) \\ = \underset{n \to \infty}{L t} \frac{1}{(1 - x)} (1 - x^{4 n - 4}) (1 + x^{2 n - 1}) \\ = \frac{1}{1 - x} (1 - 0) (1 + 0) \\ = \frac{1}{1 - x} . \end{matrix}$

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