That 1 + 2 + 3 + 4 + ... = - 1/12 Video

[beero1000]

Yay, correctness!

 

[Kharakov]

So he's basically saying things like the following are incorrect in specific cases?

\(\frac {1}{a+b} = \sum_{n=1}^\infty \frac{(-b)^{n-1}}{a^n}\)


So the case a=1 b=-2, which gives 1+2+4+8.... "=" -1 is BS, without a valid natural analytic continuation function, even if the Cauchy product of (1-2) and (1+2+4+8...) =1?



Are Cauchy products only valid for non-divergent infinite series?

 

[beero1000]

So he's basically saying things like the following are incorrect in specific cases?

\(\frac {1}{a+b} = \sum_{n=1}^\infty \frac{(-b)^{n-1}}{a^n}\)


So the case a=1 b=-2, which gives 1+2+4+8.... "=" -1 is BS, without a valid natural analytic continuation function, even if the Cauchy product of (1-2) and (1+2+4+8...) =1?



Are Cauchy products only valid for non-divergent infinite series?

No, he's saying that naive manipulations need to be justified or you can get contradictions. The natural function for a geometric series is 1/(1 - z), so it can be analytically continued everywhere except z = 1. When z = 2 you do get -1, but it's important to understand which operation was performed (e.g. it is not 1 + 2 + 4 + 8 + ... = -1), and why it was legal.

Cauchy products are defined for all infinite series, but you cannot assume they will converge and cannot blindly manipulate them as if they were convergent without justification.

 

[Artemus]

Very good video. The great  Peter Schickele always claims to be a Professor of Musical Pathology at the University of Southern North Dakota at Hoople. Perhaps they can open a Department of Mathematical Pathology to address things like the original video that made the bogus claim.

I have to remind my electromagnetics students that the fact that

\(\int_a^b \frac{1}{x} dx = \log\left(\frac{b}{a}\right)\)

gives the correct answer when a < 0 and b < 0 does not mean that the individual terms of the intermediate step

\(\log(b)-\log(a)\)

have meaning if we restrict ourselves to real numbers. It gives a good introduction to the review of analytic continuation and the principal value of the integral when a*b < 1.

 

[Shake]

Nice debunking! I mean I knew, logically, it made no sense for a sum of positive integers (indeed, all the positive integers) to sum to a negative number, but the Numberphile guys seemed to present a good case. And of course I get divergent series, etc., I just hadn't thought about it very much.

Hmm ... has anyone told the guys on Numberphile yet? To YouTube!