Question

The sum $1+\frac{2}{x}+\frac{4}{x^2}+\frac{8}{x^3}+....\left(upto\infty \right),x\ne 0,$ is finite if

• A. $\left|x\right|<2$
• B. $\left|x\right|>2$
• C. $\left|x\right|<1$
• D. $2\left|x\right|<1$

Answer 1

Answer: (B)

$\left|x\right|>2$

$1+\frac{2}{x}+\frac{4}{x^3}+\frac{8}{x^3}.......upto\infty$

It is infinite $G.P$ with Common ratio $r=\frac{2}{x}$

It's sum is infinite if $\left|r\right|<1$

i-e

$\left|\frac{2}{x}\right|<1$

$\Rightarrow \frac{2}{\left|x\right|}<1$

$\Rightarrow 2<\left|x\right|$

$\Rightarrow \left|x\right|>2$