2

I have come across the proof that all natural numbers to infinity when summed equals -1 / 12 and can not quite grasp this. Could anyone find a way to explain this in an easy understandable way for someone still in school (for that one kid who says he gets this and he's still in school, good for you).

Here's the video I saw : https://www.youtube.com/watch?v=w-I6XTVZXww

Thanks!

Emilio Novati
  • 62,675
Plus Twenty
  • 838
  • Mathologer made an excellent video about that topic on YouTube. – G. Ünther Jul 24 '17 at 08:44
  • Here: https://m.youtube.com/watch?v=jcKRGpMiVTw – G. Ünther Jul 24 '17 at 08:45
  • 1
    See whether this helps you: https://plus.maths.org/content/infinity-or-just-112 –  Jul 24 '17 at 08:46
  • Simple explanation: At its heart, mathematics is about taking ideas we are familiar with, creating a solid foundation using (preferably obvious) definitions upon which we can work, and then carefully inspecting the properties of the structure we have created. We then see what happens when we change some of our assumptions, and study the new structure, hopefully ending up with something beautiful, intuitive, useful, or all of the above. – Brevan Ellefsen Jul 24 '17 at 08:52
  • In your case, what happens is that we start with addition, and end up generalizing it to a bunch of things, such as infinite sums, integration, etc. We then find that when we relax some of our assumptions in seemingly different ways we get seemingly (somewhat) consistent results such as $\sum n = \frac{-1}{12}$. The natural question is to then ask if this generalized form of addition has any interesting properties, and to ask what a nice foundation for it is. At this point you start looking into the ways that series rearrangements, the Euler-Maclaurin Summation Formula, the $\zeta$ function, – Brevan Ellefsen Jul 24 '17 at 08:55
  • etc. relate to each other, and discover underlying patterns and structures you perhaps didn't realize existed prior. Now, whether this is beautiful is subjective - nevertheless, many people, including mathematical giants such as Cesaro, Abel (somewhat), Ramanujan, Euler, and so forth found these structures interesting and published their works on them. Since then some fields of physics have found applications to this type of summation, which is perhaps even more justification for understanding exactly what is happening. – Brevan Ellefsen Jul 24 '17 at 09:00
  • Summing up: for me, the true beauty in it is how we can derive such a seemingly nonsensical, absurd, and unintuitive result by relaxing conditions on summation in a way that feels very natural, and that we get the same result in a bunch of seemingly non-connected ways. – Brevan Ellefsen Jul 24 '17 at 09:02
  • Thank-you everyone for your thoughts. It is quite amazing how the answer defies our basic intuition we learn early school. I will leave this open for a while in case others have their 2 cents they wish to add. Thanks – Plus Twenty Jul 24 '17 at 09:17

1 Answers1

0

I think that there is no ''easy understandable way'' to explain this result to a middle/high school student.

It is really a strange result because this series does not have a limit in the usual sense, and this value is found using a more general definition of the sum of a series and a method that use the properties of an analytic function (the zeta function regularization) so that the knowledge of the complex analysis is needed to understand the result.

There are different methods to define the ''sum'' of a divergent series and the reference book for this topic is G. Hardy, Divergent series.

You can see also other questions about this topic as: Is $1+2+3+4+\cdots=-\frac{1}{12}$ the unique ''value'' of this series?

376
  • 7
  • 2
Emilio Novati
  • 62,675