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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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marked as duplicate by Zev Chonoles Dec 12 '11 at 19:33

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    $\begingroup$ the extension of $\sum z^n$ to the rest of the plane is $1/(1-z)$ which is $-1$ at $z=2$ $\endgroup$ – yoyo Dec 12 '11 at 19:34
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    $\begingroup$ Manipulated series must be convergent to render the arithmetic sensible. $\endgroup$ – user13838 Dec 12 '11 at 19:34
  • $\begingroup$ @yoyo, what do you mean by the extension of $\sum z^n$? $\endgroup$ – David Souther Dec 12 '11 at 19:52
  • $\begingroup$ @David: he meant "analytic continuation". You do have what's called a "geometric series"... $\endgroup$ – J. M. is a poor mathematician Dec 13 '11 at 1:01
  • $\begingroup$ @percusse: This series is convergent in the $2$-adic metric and indeed converges to $-1$. $\endgroup$ – Zhen Lin Dec 13 '11 at 2:54