The sum of infinite series $1-3x+5x^2-7x^3+\mathrm{.........}\infty ,$ $\left|x\right|<1$ is  equal to 2, then$x$ is:

$S=1-3x+5x^2-7x^3+\mathrm{.............}$

$xS=x-3x^2+5x^3-\mathrm{........}$

$\Rightarrow \left(1+x\right)S=1-2\left[x-x^2+x^3-\mathrm{........}\right]$

$\Rightarrow S=\frac{1+x-2x}{{\left(1+x\right)}^2}=\frac{1-x}{{\left(1+x\right)}^2}=2$          $\left[\because \left|x\right|<1\right]$

$\Rightarrow$  $2x^2+5x+1=0$ $\Rightarrow x=\left(-5\pm \sqrt{17}\right)/4$

$\Rightarrow$ $x=\left(\sqrt{17}-5\right)/4$  as    $\left|\frac{-5-\sqrt{17}}{4}\right|>1$