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Possible Duplicate:
Infinity = -1 paradox

MinutePhysics has what initially looks like a divergent series summing to -1. The youtube comments are... lacking in clarity. The argument MinutePhysics loosely makes is

$1+2+4+8+16...$

$=(1)(1+2+4+8+16...)$

$=(2-1)(1+2+4+8+16...)$

$=(2+4+8+16+32...) - (1+2+4+8+16...)$

$=-1 + (2+4+8+16+32...)-(2+4+8+16...)$

Is this proof correct, and if not, what is the error?

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    the extension of $\sum z^n$ to the rest of the plane is $1/(1-z)$ which is $-1$ at $z=2$2011-12-12
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    Manipulated series must be convergent to render the arithmetic sensible.2011-12-12
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    @yoyo, what do you mean by the extension of $\sum z^n$?2011-12-12
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    @David: he meant "analytic continuation". You do have what's called a "geometric series"...2011-12-13
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    @percusse: This series is convergent in the $2$-adic metric and indeed converges to $-1$.2011-12-13
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    @ZhenLin Yep, but I didn't want to complicate things. That's why I didn't write *the series are not convergent*.2011-12-13
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    Hmmmm... Im going to have to learn the p-adic system. All I know is that the geometric series diverges, therefore playing with the arithmetic is bound to create paradoxes.2013-02-12

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