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Why is the sum of all natural numbers $- \frac1{12}$?

I need a proof my 14 year-old classmates could understand, with minimal effort on my part ;)

I know I can prove it using zeta functions, etc. but that's too complex for them, currently they don't believe me, and it's quite annoying.

Other questions on this site aren't detailed or simple enough for my classmates to understand. I need simple methods of proving each step along the path of a proof.

minseong
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    The first step in the proof is to give a definition of $\sum_{n=1}^\infty$ that makes it work – Hagen von Eitzen May 03 '15 at 19:46
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    Elementary number theory is really not enough to cover something like this. The common approaches are via analytic continuation of the riemann zeta function or via regularization, and those are things most 14-year-olds won't be able to handle... – Hayden May 03 '15 at 19:46
  • http://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF#Heuristics – wythagoras May 03 '15 at 19:47
  • I found this video to be very simple https://www.youtube.com/watch?v=w-I6XTVZXww so maybe just show them this – JackV May 03 '15 at 19:47
  • I don't think any sane person would want to believe you given that you are teaching elementary number theory to your students. You should specify the scope and domain explicitly. – Landon Carter May 03 '15 at 19:47
  • In other words, you want us to help you put something over on your classmates? It looks like you have already fallen for it! – GEdgar May 03 '15 at 19:53
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    The problem with that video is that it uses very naive mathematics that can get you in trouble if you try to use it carelessly to "prove" other results. @JackV – Thomas Andrews May 03 '15 at 19:54
  • @ThomasAndrews I fully agree with you but that was just the best thing i could think of to explain it to 14 year olds (which to me seems like somewhat of a bad idea) – JackV May 03 '15 at 20:00
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    So you're asking "How can I explain a populist concept, which is often handwavvy explained as magic, without any actual mathematical justifications - because there are none - so my friends will think I'm cool?" for what it's worth, stick with the $\zeta(-1)$ reasoning. It's the most convincing one. – Asaf Karagila May 03 '15 at 20:01
  • @AsafKaragila No, it's not so that my friends will think I'm cool, I'm not a jerk, it's so that I can stop feeling frustrated by their refusal of this fact. Really, I know someone will say: "Oh, you don't really understand it yourself if you can't explain it to them"- maybe their right, but at least I know it's true. And I hate hand-waving. – minseong May 03 '15 at 20:09
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    No, it's nonsense. Being able to understand something, and being able to explain something (to a layman, to a five year old, to your grandmother, or to the recently deceased plant in your garden) are two different abilities which do not coincide. Just look at how many mathematicians do such a... crappy... work explaining their own work. Do you think they don't understand it? If you can't explain it in "simple terms", it's because this is not a simple proof, and it abuses a lot of definitions in its path. That's a *good* thing. – Asaf Karagila May 03 '15 at 20:11

1 Answers1

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Commments, before it is closed... $$ \zeta(-1) = -\frac{1}{12}\qquad\mathbf{TRUE} $$ $$ 1+2+3+\cdots = -\frac{1}{12}\qquad\mathbf{FALSE} $$ $$ 1+2+3+\cdots = \zeta(-1)\qquad\mathbf{FALSE} $$

Now the one in the book seen in the infamous video is $$ 1+2+3+\cdots \rightarrow -\frac{1}{12} $$ which may or may not be true, depending on how you define that $\rightarrow$ symbol.

GEdgar
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