If: $$S=1+2+4+8+....+2^n +...$$ So we get $$2S=2+4+8+...$$ $$2S+1=1+2+4+8+16...$$ $$2S+1=S$$ $$2S-S=-1$$ $$S=-1$$
Is there error, and if there's, why? I want athletic explanation.
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1$S\notin \Bbb R$ that's the problem... – Surb Jul 07 '15 at 21:02
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3The series doesn't converge. Don't play with $\infty$ – Teoc Jul 07 '15 at 21:02
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@Surb , Can you give me you're explanation? – TEXS Jul 07 '15 at 21:03
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4Jeez, can't anyone search the site before posting this for the $(1+2+4+\cdots)$th time? – Zev Chonoles Jul 07 '15 at 21:03
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4But most importantly: what's an athletic explanation? – Ant Jul 07 '15 at 21:04
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1$S$ is not a real number! You can't add, subtract and multiply arbitrary things and expect them to act like real numbers. – balddraz Jul 07 '15 at 21:04
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1So, you are concluding $2\times \infty+1=\infty\implies \infty = -1$? Interesting. – Mark Viola Jul 07 '15 at 21:05
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In other words, this is a another illustration of why treating $\infty$ as a real number leads to contradictions. – Simon S Jul 07 '15 at 21:06
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1@ZevChonoles Now that was unboundedly, "series-ly" hilarious! – Mark Viola Jul 07 '15 at 21:08
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1@Ant That is a good one! – Mark Viola Jul 07 '15 at 21:09
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@Ant: An athletic explanation supports the conclusion. – André Nicolas Jul 07 '15 at 21:23
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1@AndréNicolas as opposed to an explanation that gives the right answer? Never heard of this idiom before :) – Ant Jul 07 '15 at 21:24
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@ant I never heard the idiom before either, but Andre was referring to American usage for https://en.wikipedia.org/wiki/Jockstrap – Will Jagy Jul 08 '15 at 01:44
1 Answers
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Here is your athletic explanation, as requested. Your series are divergent in the real numbers and have no meaning.
On the other hand, the thing is convergent in the $2$-adic numbers $\mathbb Q_2,$ and the sum really is $-1.$ https://en.wikipedia.org/wiki/P-adic_number
This is problem 21 on page 20 of Gouvea, http://www.amazon.com/p-adic-Numbers-An-Introduction-Universitext/dp/3540629114
Will Jagy
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