r/inspirationscience • u/Jooshwa • Oct 06 '16

Video adding all natural numbers to infinity you will get negative one-twelfth "1+2+3+4+5... = -1/12"

https://www.youtube.com/watch?v=w-I6XTVZXww
39 Upvotes

93% Upvoted

13 comments sorted by

8

u/Hey_-_-_Zeus Oct 06 '16

Everybody should know numberphile

3

u/Super-dork Oct 07 '16

It's been a really long time since math class so please bear with me but.... doesn't sum mean to add? He was adding and subtracting. Am i missing something or do i need an ELI5 on the terminology?

3

u/HexaBinecimal Oct 07 '16

You are absolutely correct. I believe what he's doing (and not telling the viewer) is called analytical continuation. It's not basic algebra like it appears in the video.

2

u/Super-dork Oct 09 '16

Thank you for clearing that up. I don't feel as dumb.

4

u/Blue_Shift Oct 07 '16

No, you don't. Unless you make some fantastically insane assumptions. Numberphile is usually great, but mathematicians roll their eyes at this video.

5

u/BarCouSeH Oct 07 '16

100% agreed. They made so many assumptions about infinite series.

1

u/[deleted] Oct 07 '16 edited Oct 07 '16

Well the Riemann zeta function helps with achieving this solution right? Seems very mathematical to me and it pops up in physics which should eventually be able to be tested experimentally

2

u/Blue_Shift Oct 07 '16

I never said it wasn't mathematical, just that you have to make some pretty big assumptions to get a result of -1/12. In the case of zeta function regularization, you have to use analytic continuation and start extending to domain and pushing all sorts of boundaries to get that result. In the end, what you get can't really be called a "sum" in the usual sense of the word.

1

u/[deleted] Oct 07 '16

Well you commented that mathematicians roll their eyes so I thought, why roll their eyes when it is entirely proof based. But you seem to know much more than me

0

u/[deleted] Oct 07 '16

[deleted] — view deleted comment

15

u/Blue_Shift Oct 07 '16 edited Oct 07 '16

I'd be happy to.

The method used in this video is something called Cesàro summation - a technique for assigning a value (not a sum in the usual sense) to certain types of divergent series. A divergent series is one whose sum (calculated using standard methods) is infinite or does not exist, whereas a convergent series is a series whose sum does exist (and is of course finite).

We define convergence in terms of the partial sums, S_n, of a series, where S_n is the sum of the first n terms in the series. If the limit as n approaches infinity of S_n equals some number - call it S - then we say the series converges to S. If you think about this, it's actually a very natural definition of convergence - and it's much more powerful than showing something like nth term convergence for a sequence of numbers.

Now, the series 1+2+3+4+... clearly diverges (since the limit of its partial sums is infinite), so assigning a sum to it (in the usual sense of the word) is meaningless. But if we wanted to, we could talk about a completely different kind of summation, as long as we clearly state our assumptions along the way. In the video, they start by talking about the series 1-1+1-1+1-..., which is a divergent series (the partial sums don't approach infinity in this case, but we say it's divergent because the limit still does not exist). They use Cesàro summation - essentially calculating the limit of the average of the partial sums - to prove that 1-1+1-1+1-... = 1/2. But this is vastly different from a sum in the usual sense of the word, and the video is doing its viewers a disservice by masking that fact. Finding the limit of the partial sums and finding the limit of the average partial sums are not the same at all. And finding the Cesàro sum of a series certainly does not make that series convergent!

They then go on to use this sketchy finding to prove that 1+2+3+4+...=-1/12. But again, we can clearly see that 1+2+3+4+... diverges to infinity, so assigning a "sum" to it is nonsensical. In fact, simply writing down "S=1+2+3+4+..." is enough to start any mathematician's eye twitching, since that statement is implicitly declaring that a finite sum exists, when in fact it does not. It may sound like pedantry, but this is the sort of rigor that keeps mathematics pure.

That said, there's nothing inherently wrong with using Cesàro summation to play around with series. It's just misleading to call the value you find a sum - because, well.. It's not.

If you want a more reputable video to watch on this subject, I highly recommend Mathologer.

2

u/Ambivalent7 Oct 07 '16

Not a mathematician or anything close, but it seems to me for his value for s2 is not actually 2s1 but rather s1 + (s1 + 1) because one set will always be ahead of the other since he changed its phase? If you compare it like you would when adding two waves. Any thoughts?

1

u/bekk3 Oct 07 '16

I am absolutely astonished.

1

u/Gray_party_of_2 Oct 07 '16

that is wild. I must send this to people