Standard Generating function defined as :
$G(x)=a_{0}x^{0}+a_{1}x^{1}+a_{2}x^{2}+a_{3}x^{3}+a_{4}x^{4}+a_{5}x^{5}+a_{6}x^{6}+........$
$G(x)=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+a_{4}x^{4}+a_{5}x^{5}+a_{6}x^{6}+........$--------------->$(1)$
where $a_{0},a_{1},a_{2},a_{2},a_{3},a_{4},a_{5},..........$ are generating sequence
Given that generating sequence are $0,1,−2,4,−8,16,−32,64,.....$
$G(x)=0+1.x-2x^{2}+4x^{3}-8x^{4}+16x^{5}-32x^{6}+64x^{7}-..........$ $[$From the equation$(1)]$
$G(x)=x - 2x^2 + 2^2x^3 -2^3x^4 + 2^4x^5 - 2^5x^6 + 2^6x^7-........$----------->$(2)$
This is a infinite GP where $a = x$ and $r = -2x$
$\large S_\infty = \frac{a}{1-r} =$$\Large \frac{x}{1+2x}$
So$,G(x)=\frac{x}{1+2x}$