I can't quite wrap my head around this.
Given the formula
$(1-x)(1+x+x^2+...) = 1$
It seems clear to me why this is true. All the x terms cancel out and we are left with one. And this is clearly true for all values of x.
However what I can't figure out is
$\displaystyle\sum_{i=0}^\infty x^i = \frac{1}{1-x}$
If x is something like 2, then
$\displaystyle\frac{1}{1-2} = -1$
But $\displaystyle\sum_{i=0}^\infty 2^i$ is the sum of infinitely many positive numbers. How is it possible that they are equal?
The series $\sum_{i=0}^\infty x^i$ converges if $-1<x<1$ and otherwise diverges. If it converges, then it is $1/(1-x)$.