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I do understand and am able to reproduce the steps to proove that $$1+2+3+...=-\frac{1}{12}$$ as, for example, shown in the Numberphile YouTube video.

I can proove it, but I can't understand it. My brain is unwilling to accept that a series $a = \sum_{i = 1}^{n}i$ with $n \rightarrow \infty$ evaluates to $-\frac{1}{12}$.

So what does it mean? I have two ideas:

  • Human logic and math is not coherent, so this is an edge case that is not covered.
  • The logic of the universe is not coherent, e.g. a bug / incomplete feature of reality.
  • $-\frac1{12}$ has some deeper meaning we do not know yet.

None of these points seem believable, so I am thinking that my understanding of this kind of Math might be too limited. Maybe you can enlight me?

Also please edit this question if you see mistakes, I am fairly new and definitley not a mathematician. I only know LaTex, that's all.

Hanul Jeon
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Markus Appel
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    it is illogical. the answer LOGICALLY HAS TO BE: of a high magnitude, positive and integer. fortunately, to my understanding it is not widely used – Alex Robinson Jan 17 '18 at 10:34
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    Or, your brain is right and that Numberphile video is garbage. It's not a true equality. The LHS and RHS are inherently connected, but not equal in the usual sense. That divergent sum can be regularised to that value (unique, if we abide by certain rules). By the way, take a look at this other video from Numberphile, it's a sort of amendment. – Vincenzo Oliva Jan 17 '18 at 10:37
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    See https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_⋯#Summability – lhf Jan 17 '18 at 10:37
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    The left-hand side just doesn't mean what it looks like it means. It's a misleading use of notation. – Qiaochu Yuan Jan 17 '18 at 10:38
  • @QiaochuYuan Then what does it mean? – Markus Appel Jan 17 '18 at 10:39
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    See Wiki's entry likned above: "the methods of zeta function regularization and Ramanujan summation assign the series a value of $\dfrac {1} {12}$, which is expressed by a famous formula: ${\displaystyle 1+2+3+4+\cdots =-{\frac {1}{12}},}$ where the left-hand side has to be interpreted as being the value obtained by using one of the aforementioned summation methods and not as the sum of an infinite series in its usual meaning." – Mauro ALLEGRANZA Jan 17 '18 at 10:42
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    read this Terence Tao's aritcle for some insight. – achille hui Jan 17 '18 at 10:43
  • Mathologer on YouTube offers a completely logical and correct evaluation on this sum. I recommend watching the full video. – Landuros Jan 17 '18 at 10:46
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    In conclusion, the answer is a modification of yout third bullet: "infinite summation has some different meaning that I do not know yet." – Mauro ALLEGRANZA Jan 17 '18 at 10:47
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    Obviously, the series $1 + 2 + \ldots$ does not converge. If it converged, it'd do it to a number in $\mathbb{Z}$ because $(\mathbb{Z} , +)$ is a group and you're adding whole numbers, so the statement $1 + 2 + \ldots = - \frac{1}{12}$ is clearly false. – joseabp91 Jan 17 '18 at 11:05
  • this video explain in detail what it means. – Masacroso Jan 17 '18 at 11:26
  • Sadly, serious videos do not attract many viewers. This is the reason that , for example , there are not many videos about the non-existence of UFO's. – Peter Jan 17 '18 at 11:53
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    @Peter oh UFO's do exist, lots of them. But none of them are secret government projects or alien spaceships. – Markus Appel Jan 17 '18 at 12:31
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    @joseabp91 Not necessarily. $(\mathbb{Q}, +)$ is also a group but an infinite sum might converge to something that is not in $\mathbb{Q}$. – badjohn Jan 17 '18 at 13:16
  • @badjohn thank you by your true observation. Anyway, my first statement is true and it admits an easy proof. – joseabp91 Jan 17 '18 at 13:33
  • @joseabp91 Of course, in this case, the sum does not converge in any usual sense. My point was just that you cannot assume that properties of finite sums extrapolate to infinite ones. For example, finite sums in $\mathbb{R}$ are commutative but infinite ones might not be. – badjohn Jan 17 '18 at 14:09
  • @badjohn I agree your statement. My second statement is a mistake. Thank you. – joseabp91 Jan 17 '18 at 14:15

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