I have nothing but the highest respect for Euler, but he did not do rigorous math the way we mean that term today. So Euler's non-proof is not even faintly convincing to me in terms of the modern meaning of the convergence of a series. In terms of modern math notation, the equation (*) 1+2+3+... = -1/12 is true only if you change the meaning of the notation. On the other hand, Terry Tao (for whom I also have nothing but the greatest respect, and who uses modern math notation) has a lot to say about (*) here: < http://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoul... >. --Dan On Apr 16, 2014, at 3:45 PM, Mike Stay <metaweta@gmail.com> wrote:
Well, it's Euler's proof, so not entirely unconvincing. Euler just had the exquisite taste to know which series would have analytic continuations even though the theory hadn't been developed yet.
On Wed, Apr 16, 2014 at 4:12 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
The reason that people attribute -1/12 to this sum is that it can be considered to be Zeta[-1], where we define Zeta[z] (z > 1 for convergence) to be:
1/1^z + 1/2^z + 1/3^z + 1/4^z + ...
Then Zeta[] has a unique analytic continuation to a meromorphic function over the complex plane (with a single pole at 1), and Zeta[-1] = -1/12.
That's the only reasonable evidence I've seen in favour of this. The popular `numberphile' video giving a purported `proof' using elementary methods is unconvincing hand-wavery.