<s>[INST] Prove $ 1 + 2 + 4 + 8 + \dots = -1$ 
Possible Duplicate:
Infinity = -1 paradox 

I was told by a friend that $1 + 2 + 4 + 8 + \dots$ equaled negative one. When I asked for an explanation, he said:

Do I have to?
Okay so, Let $x = 1+2+4+8+\dots$
$2x-x=x$
$2(1+2+4+8+\dots) - (1+2+4+8+\dots) = (1+2+4+8+\dots)$
Therefore, $(2+4+8+16+\dots) + (-1-2-4-8+\dots) = (1+2+4+8+\dots)$. Now $-2$ and $2$, $-4$ and $4$, $-8$ and $8$ and so on, cancel out, and the only thing left is $-1$.
Therefore, $1+2+4+8+\dots = -1$.

I feel that this conclusion is not right, but I cannot express it. Can anyone tell if this proof is wrong, and if it is, how it is wrong?
 [/INST] Look at Calculus textbook, undergraduate level.  When we treat infinite sum, we cannot change the order to compute.
Example. $1-1+1-\cdots$
$$(1-1)+(1-1)+\cdots=0+0+\cdots=0$$
$$1+(-1+1)+(-1+1)+\cdots=1.$$
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