Question

If the sum of the series $1+\frac{2}{x}+\frac{4}{x^2}+\frac{8}{x^3}+......\infty$ is a finite number, then:

• A. $x>2$
• B. $x>-2$
• C. $x>\frac{1}{2}$
• D. None of these

Answer 1

The correct option is A

$x>2$

Explanation for the correct option.

In the given G.P. series $1+\frac{2}{x}+\frac{4}{x^2}+\frac{8}{x^3}+......\infty$, $a=1\text{and}r=\frac{2}{x}$

Sum of infinite terms of G.P, $\mathit{S}\mathbf{=}\frac{\mathbf{a}}{\mathbf{1}\mathbf{-}\mathbf{r}}$

$\begin{array}{rcl}& \Rightarrow & S=\frac{1}{1-{\displaystyle \frac{2}{x}}}\\ & =& \frac{x}{x-2}\\ & & \end{array}$

Now, $S$ will be finite if $x-2>0$. Therefore, $x>2$.

Hence, option A is correct.