Question

Let $f\left(x\right)=x^2+x^4+x^6+x^8+...\infty$ for all real $x$ such that the sum converges. Number of real $x$ for which the equation $f\left(x\right)-x=0$ holds, is

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is D $3$

$f\left(x\right)=x^2+x^4+....\infty =x^2\left(\frac{1}{1-x^2}\right)$ (By G.P. Summation)
$f\left(x\right)-x=0$
$\frac{x^2}{1-x^2}-x=0\Rightarrow x^2=x(1-x^2)$
$\Rightarrow x^2+x^3-x=0\Rightarrow x(x^2+x-1)=0$
$x=0,\frac{-1\pm \sqrt{5}}{2}$
There are 3 values