Can the Infinite Sum of Natural Numbers Really Equal -1/12?

[J.J.T.]

I was some youtube videos and i got sucked into this channel called "numberphile". They were talking about infinite sets. In particular the set that is the sum of all natural numbers. Through some creative algebra they demonstrate the proof. Somehow the set that is equal to the sum of all natural numbers :1+2+3+4+5+6+7+... is equivalent to -1/12. The algebra is easy enough to follow that a high school student could keep up quite easily. But intuitively I just can't accept it. They say that whenever this set is encountered in their mathematics by simply substituting -1/12 the math is accurate every single time. Anyone here a total math whiz that can explain this in a way that makes sense intuitively rather than just "look here's the proof, we know its insane but it works!"?

I'm pretty good at math, but I was away from math for awhile and i no longer have that mathematical "intuition" that might've helped me understand the concept underneath the proof.

 

[HallsofIvy]

The proof only works if you are not careful about the "details"! In particular the "algebra" used only works on a convergent series and it is easy to show that this sequence is not convergent.

 

[micromass]

"look here's the proof, we know its insane but it works!"?

Well, it looks insane but it doesn't work. The video is highly misleading. Any honest mathematician will say that $1+2+3+4+5+...$ equals infinity (or more correctly: diverges to infinity). This is the standard interpretation of that infinite sum.

But some mathematicians have found ways to still give a number to infinite sums. This should not be interpreted as the total sum, it should just be interpreted as some number you get when you follow a certain procedure. The number given to $1+2+3+4+5+...$ is indeed $-1/12$. Again, this does not mean that the total sum equals $-1/12$, it means that if we follow a certain procedure with that sum (like the procedure in the video), we get $-1/12$.

 

[WWGD]

Like others have said, the '=' sign is assumed, by default, that the sum _converges_ to that value, when it does not. The '=' is used with another intended meaning.

 

[Nikhil981888]

Hey,
-1/12 is not the perfect sum of this series but it's called partial sum of the series.
Leaving it Checkout my forum
https://www.physicsforums.com/threads/1-2-3-4-5-6-7.854564/

 

[HallsofIvy]

Hey,
-1/12 is not the perfect sum of this series but it's called partial sum of the series.
Leaving it Checkout my forum
https://www.physicsforums.com/threads/1-2-3-4-5-6-7.854564/

NO, it isn't. An infinite sum, whether it converges or not, has an infinite number of "partial sums", not just one. The finite sum \sum_{i=0}^n a_i is a partial sum of sum \sum_{i=0}^\infty a_i for all n.

 

[Mark44]

Hey,
-1/12 is not the perfect sum of this series but it's called partial sum of the series.

I don't think you understand what "partial sum" means. For an infinite series to converge, its sequence of partial sums must converge.

In the series of this thread, 1 + 2 + 3 + ... +, here's the sequence of partial sums:
$S_n = \{1, 3, 6, 10, 15, \dots \}$
The farther you go in the sequence of partial sums, the bigger the term in the sequence gets.