We are going to use a trick called telescoping.
(1 - x)(1 + x + x²) can be split the positive terms and the negative terms.
Positive: 1 + x + x²
Negative: - x - x² - x³
The two of the positive and negative terms cancel out and we are left with just 1 - x³.
If we have a number x between -1 and 1, as the powers increase towards infinity the number gets closer to zero. Dividing both sides by (1 - x) tells us the sum will be equal to 1/(1 - x).
For example, if x = ½, the sum 1 + 1/2 + 1/4 + 1/8 + ... = 1/(1 - ½) = 1/½ = 2.
if x = -½, the sum 1 - ½ + 1/4 - 1/8 + ... = 1/(1 - (-½)) = 1/(3/2) = 2/3.
There are many other methods used in infinite sums, but this is one of the most basic.
