Converting Summation to Expression How does the summation break down from
$$\displaystyle\sum_{n \geq 0} (x + x^2) ^ n$$
to
$$\frac1{1 - x - x^2} $$
per this answer?

Assuming $$|x+x^2|\lt 1$$
$$s=\displaystyle\sum_{n \geq 0} (x + x^2) ^ n$$
Notice that
$$\quad \\\\\\\ \  \ \quad s=1+(x+x^2)+(x+x^2)^2+(x+x^2)^3+\cdots$$
$$s(x+x^2)=(x+x^2)+(x+x^2)^2+(x+x^2)^3+\cdots$$
$$s-s(x+x^2)=s(1-(x+x^2))=1$$
$$\Rightarrow s=\frac{1}{1-(x+x^2)}=\frac{1}{1-x-x^2}$$
$$\Rightarrow \displaystyle\sum_{n \geq 0} (x + x^2) ^ n=\frac{1}{1-x-x^2}$$
