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 the complete solution of x will be -
[UPSEAT 2002]

• A. x > 2
• B. x < -2
• C. (x < -2) U (x > 2)
• D. None of these

Answer 1

The correct option is C (x < -2) U (x > 2)

Common ratio (r) = $\frac{2}{x}$
For the sum of an infinite G. P. to be a finite no. its common ratio should lie between (-1, 1)

So, $-1<\frac{2}{x}<1$
$\Rightarrow \frac{2}{x}<1$
$\Rightarrow \frac{2}{x}-1<0$
$or\frac{2-x}{x}<0$
$\Rightarrow xϵ(-\infty ,0)\cup (2,\infty )$
Similarly, $\frac{2}{x}>-1$
$\frac{2}{x}>1>0$
$=\frac{2+x}{2}>0$
$\Rightarrow -\infty ,-2)\cup (2,\infty )$
On taking intersection we get,
$-\infty ,-2)\cup (2,\infty )$