How Does Summing Infinite Integers Equal Negative One-Twelfth?

[atyy]

I agree that one should encourage the challenge of ideas, but it seems like a lot of background is needed to be able to present this challenge in a coherent way; maybe if it is a really advanced class, but do you think you can use concepts like analytic continuation, at the high school level?

Maybe if you have someone particularly good at presenting the general ideas of convergence, continuation, but this is rare. Sorry if my post came off as aggressive.

But historically the formula was discovered without analytic continuation. Is it so terrible to present it analogous to the way old quantum theory is still taught before quantum mechanics? I'm pretty sure all students, even the worst, know that the "=" sign is not the usual one.

 

[atyy]

Here is another interesting comment on the formula and the place of "formal derivations" by Atiyah in his essay "How Research is Carried Out" (https://books.google.com/books?id=YJ0cZwxLECAC&printsec=frontcover#v=onepage&q&f=false, p213)

"The next point I want to mention is the distinction between formalism and rigour. Again this is a dichotomy in mathematics which has a long history. In the formal kind of mathematics you do things without worrying too much about precisely what they mean as long as they give the right answer. You might say this only happens in applied mathematics but that is not quite correct; I think that it also occurs in pure mathematics. Of course, historically, there are famous examples of the formal approach. The most famous practitioner, no doubt, was Euler, who, as you know, produced many magnificent formulae. He "evaluated" such wildly divergent series like $\sum_{n=1}^{\infty} n = -1/12$. It was only a century or so later that precise meaning could be attached to such formulae."

 

[Michael David]

At about 1:11:20 ish he starts talking about 1+2+3+...

I found this quite helpful