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The Sum of All the Natural Numbers 
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#1
Jan1014, 05:31 PM

P: 68

Hi lovely people,
I recently came across a video http://www.youtube.com/watch?v=wI6XTVZXww that said if you add all of the natural numbers from 1 to infinity, the answer is... What do you think it is? Infinity or something like that? They said it was 1/12. I watched the proof but I don't understand the logic behind it because if you add positive numbers together, how can you get a negative? And if you're adding whole numbers together, how can you get a fraction (not like 5=5/1, you know what I mean)? I heard them say that it's essential to String Theory and all 26 dimensions coming out. I assume they meant bosonic string theory? Because that's the only subset of string theory I know that has 26 dimensions, I'm pretty sure the original has 10. I'm hoping that someone can shed some light on this so called 'astounding result'. Thanks in advance, AlfieD 


#2
Jan1014, 06:20 PM

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P: 12,081

This is not the sum of all natural numbers. He uses formulas in a region where they do not apply.
As a comparison: "all odd numbers between 2 and 8 are prime" is true (as 3, 5 and 7 are prime) and easy to prove, but that does not mean 1 or 9 would be prime as well because they are not in the region where the statement is usable. 


#3
Jan1014, 06:24 PM

P: 68




#4
Jan1014, 06:33 PM

P: 1,305

The Sum of All the Natural Numbers



#5
Jan1014, 06:34 PM

Mentor
P: 12,081

@AlfieD: Yeah sure, but it is wrong. Entertaining, but wrong.
With incorrect manipulations of infinite sums, you can prove anything. As an example, consider the sum 1/2  1/3 + 1/4  1/5 + 1/6  1/7 +... Clearly the sum is positive everywhere and increases every two steps, so it is larger than zero. 1/2  1/3 + 1/4  1/5 + 1/6  1/7 + ... > 0 Let's rearrange it a bit:  1/3  1/5 + 1/2  1/7  1/9 + 1/4 + ... Now 1/3  1/5 = 8/15 and 8/15>8/16=1/2, and in the same way 1/7  1/9 = 16/63 > 16/64 = 1/4 and so on. Therefore, this sum is clearly negative.  1/3  1/5 + 1/2  1/7  1/9 + 1/4 + ... < 0 How can the same sum be larger and smaller than zero at the same time? Well, the answer is bad mathematics  this rearrangement is not valid, it changes the value of the sum. The same is true for the steps made in the video  they are just not valid mathematics. @1MileCrash: Yes I watched it. Maybe "formulas" is not the best word, let's say "calculations". 


#6
Jan1014, 06:40 PM

P: 1,305

I'm not very convinced there there was anything wrong with the work he did, but I will look at what you said more closely. 


#7
Jan1014, 06:44 PM

P: 68




#8
Jan1014, 06:46 PM

P: 68




#9
Jan1014, 07:02 PM

P: 572

http://www.youtube.com/watch?v=Ed9m...ature=youtu.be
Is their other 'proof' video, but it's still incorrect. The first video is just mathematical hand waving. They're misusing the mathematics. 


#10
Jan1014, 07:08 PM

P: 460

He also rearranges the sums for his second sum (I thinkI have watched the video, but not today). Since an infinite sum is usually defined as the limit of the sequence of partial sums, this is not acceptable. This is not something that I know much about, but I have been told that the truth of this statement is important to some conclusions in math and physics, but his "proof" is extremely flawed. 


#11
Jan1014, 07:22 PM

P: 1,305




#12
Jan1014, 07:27 PM

P: 68

Can I just ask whether the quarrel anyone has is with this particular proof and not the actual answer itself? So, are you fine with it being 1/12, but you just think that the proof is highly floored. If so, do you know of any better proofs?



#13
Jan1014, 07:34 PM

P: 572

It's only 1/12 when used in the context of ζ(1). Suggesting that the infinite sum 1+2+3+4+5... is 1/12 is just wrong, it's an abuse of notation.
Someone correct me if I'm wrong, I'm working off a limited knowledge base here. 


#14
Jan1014, 07:52 PM

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P: 12,081

As another example: Assume 1+2+3+.... = 1/12. Then clearly 0+1+2+3+.... = 1/12. Taking the difference: 1+1+1+... = 0. In the same way, 0+1+1+1+... = 0. Taking the difference again, 1+0+0+..=0 1=0 Wait... no. 


#15
Jan1014, 07:56 PM

P: 68

Ummm, if you watch the video the guy shows you the sum inside a book entitled String Theory. Listen closely to what he says as well. I'm pretty sure it's near the start of the video. 


#16
Jan1014, 08:12 PM

P: 615

Edit: Perhaps that's what the idea is, but in the video all I see is mathematical crackpottery. It hurts my eyes. 


#17
Jan1014, 09:55 PM

P: 460

The formula is not wrong when it is viewed properly, but the video is ambiguous and uses false mathematics for a "proof" that is meaningless. 


#18
Jan1014, 11:25 PM

Sci Advisor
P: 8,796

Although the proof is wrong, some mathematicians have savoured it. In particular, Atiyah writes that the erroneous summation was Euler's, and that it was only in relatively recent times that we understand what Euler's intuition was pointing towards.
Another good fun source that discusses the history of this equation is John Baez's My Favorite Numbers: 24 http://math.ucr.edu/home/baez/numbers/24.pdf http://www.youtube.com/watch?v=vzjbRhYjELo For Atiyah's comment see his article "How research is carried out" in http://books.google.com/books?id=YJ0...gbs_navlinks_s (p213). 


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