1. Tardigrade
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  3. Mathematics
  4. Let f ( x )= x 2+ x 4+ x 6+ x 8+ ldots ldots . . . ∞ for all real x such that the sum converges. Number of real x for which the equation f ( x )- x =0 holds, is

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

Sequences and Series

Solution:

$ f(x)=\frac{ x ^2}{1- x ^2}$ when $| x |<1$
$\therefore \frac{ x ^2}{1- x ^2}= x \Rightarrow x =0$
or $x =1- x ^2$
$x^2+x-1=0 \Rightarrow x=\frac{-1+\sqrt{5}}{2}, \frac{-1-\sqrt{5}}{2}$ is to be rejected