First slide

Sigma operation series

Question

The sum of series $\frac{x}{1 - x^{2}} + \frac{x^{2}}{1 - x^{4}} + \frac{x^{4}}{1 - x^{8}} + \dots$ to infinite terms, if $f | x | < 1 is$

Moderate

A

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

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

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

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

Solution

We have, $\begin{array}{l} t_{n} = \frac{x^{2^{n - 1}}}{1 - x^{2^{n - 1}}} = \frac{1 + x^{2^{n - 1}} - 1}{(1 + x^{2^{n - 1}}) (1 - x^{2^{n - 1}})} \\ = \frac{1}{1 - x^{2^{n - 1}}} - \frac{1}{1 - x^{2^{n}}} \end{array}$
Therefore, $S_{n} = \sum_{n = 1}^{n} t_{n}$
$\begin{array}{l} = [(\frac{1}{1 - x} - \frac{1}{1 - x^{2}}) + (\frac{1}{1 - x^{2}} - \frac{1}{1 - x^{4}}) + \dots + (\frac{1}{1 - x^{2^{n - 1}}} - \frac{1}{1 - x^{2^{n}}})] \\ = \frac{1}{1 - x} - \frac{1}{1 - x^{2^{n}}} \end{array}$

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