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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?

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    You'll want to see [t$h$is](http://math.stackexchange.com/questions/29023)...2011-10-18

1 Answers 1

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The series $\sum_{i=0}^\infty x^i$ converges if $-1 and otherwise diverges. If it converges, then it is $1/(1-x)$.

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    One might add that OP's first formula is **not** true for all values of $x$.2011-10-18