Converting Summation to Expression

Question

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?

Answer 1

It follows from the fact that you have a geometric series.

In general (for geometric series), for $|r|<1$, we have that $\sum_{i=1}^{\infty}a_i=\frac{a}{1-r}$, where $r$ is the common ratio and $a$ is the first term of the series.

In your case, we have that $r=x+x^2$, and $a=(x+x^2)^0=1$

Answer 2

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

$$s-s(x+x^2)=s(1-(x+x^2))=1$$ $$\Rightarrow s=\frac{1}{1-(x+x^2)}=\frac{1}{1-x-x^2}$$

$$\Rightarrow \displaystyle\sum_{n \geq 0} (x + x^2) ^ n=\frac{1}{1-x-x^2}$$