The sum of all natural numbers = -1/12

[GaryProtein]

Yes, I'm talking about that same Gradi's Sequence. I don't think it can be used in the proof of the original video precisely because it is not a sum of all natural numbers. The two premise formulae, S1 and S2, upon which the "proof" in the original video is based, IMO, is the problem. The conclusion, or proof, is based on a faulty premise. Arbitrarily deciding to average the two results of the sequence to get 1/2 is flawed.

I also think S2 is flawed because there is no requirement that all natural numbers have to be added in sequence. If you took 2+3+127865+99+872...and never repeated a number and then applied his trick of shifting the numbers over with a different randomly ordered sequence, the result would be totally different and not result in the 1/4 solution.

I'm left with the obvious analogy of adding a whole bunch of peas and never getting to a final sum, much less one that is a negative fraction and not itself a natural number.

Agreed on all counts.

There are things you cannot do with zero and infinity because they break the rules of mathematics.

 

[ack]

Yes, I'm talking about that same Gradi's Sequence. I don't think it can be used in the proof of the original video precisely because it is not a sum of all natural numbers.

Fortunately, math does not work the way you think: you can't use X to prove X. Mathematical proofs work by building on top of other established proofs, as applicable. His approach in the first video is not wrong, it's just that it's based on [sort of] half the truth, or the truth as it fits his needs (and again, these are not some silly videos with stupid, dirty tricks of divisions by zero, etc). The second video (which also references the John Baez PDF I linked to) offers a more solid proof that has stood the test of time for centuries as far as I know, and I think it would probably be beyond anyone's ability to refute it, on this forum at least.

This isn't an easy problem to deal with and mathematicians have been grappling with it since Euler and later Reimann offered their more advanced (and so far unshaken, as far as I know) proof in the 18th and 19th centuries. I think everyone agrees the result, -1/12, is completely counter intuitive, outrageous, silly perhaps, possibly flat-out mind-boggling, but as I said earlier, my biggest problem is that it's actively used in Physics and String Theory to explain things. That, to me, is mind-boggling, but I wouldn't have a problem if anyone were to say mathematicians and physicists may well be building on top of fallacies, for centuries now. But then, prove it.

 

[ack]

I've been reading up on the Riemann zeta function (2nd video, PDF), and came across the following on http://en.wikipedia.org/wiki/Riemann_zeta_function

Via analytic continuation, one can show that

gives a way to assign a finite result to the divergent series 1 + 2 + 3 + 4 + · · ·, which can be useful in certain contexts such as string theory.[SUP][4]
[/SUP]

Mind boggling to say the least. But it looks like the key here may be "certain contexts".

 

[ack]

Here's video #4 covering Riemann's zeta function (the formula is really easy to understand); watch it till about 11:30, and the area we are interested in starts at around 10:20 : z(-1) = 1+2+3+4+5+... = -1/12 and they flash the formulas that calculate real results for a given zeta value; he also claims they have calculated and verified this -1/12 result using computers (the formulas flashed on the screen around 11:30 are those in the wiki links, and are very thick to try to confirm).