$\frac{1}{1 + x} + \frac{2}{1 + x^{2}} + \frac{4}{1 + x^{4}} + \frac{8}{1 + x^{8}} + \ldots\ldots$.

Question

$\frac{1}{1 + x} + \frac{2}{1 + x^{2}} + \frac{4}{1 + x^{4}} + \frac{8}{1 + x^{8}} + \ldots\ldots$. upto ' n ' terms is equal to

Accepted Answer

Sol. [Key Concept: $\sum\ \frac{2^{k}}{1 + x^{2^{k}}}$ telescopes using $\frac{1}{1 - x} - \frac{2^{n}}{1 - x^{2^{n}}}$.]$$- \frac{1}{1 - x} + \left( \frac{1}{1 + x} + \frac{1}{1 - x} ight) + \frac{2}{1 + x^{2}} + \frac{4}{1 + x^{4}} + \ldots. + \frac{2^{n - 1}}{1 + x^{2^{n - 1}}}$$$$\begin{matrix} & \ = - \frac{1}{1 - x} + \left( \frac{2}{1 - x^{2}} + \frac{2}{1 + x^{2}} ight) + \frac{4}{1 + x^{4}} + \ldots + \frac{2^{n - 1}}{1 + x^{2^{n - 1}}} \ & \ = - \frac{1}{1 - x} + \frac{2^{n}}{1 - x^{2^{n}}} \end{matrix}$$

Question

Options

Solution

More Progression (Sequence and Series) Questions