Last time i thought i'd been careful enough, not to write 1+2+3... = -1/12 directly, talking "to (pretend to) sum" it up. I wasn't. Now, after a nice and intense discussion with  (and a little reading) i'd like to share with you what came out of it.

Nowadays it's only mildly surprising that an infinite sum, or series can yield a finite result. Thanks to Baron Augustin-Louis Cauchy, we now teach clear criteria to students who are about the age he was, when he sanitized the mess that plagued algebraists before him. So that's the boring stuff: Made by a genius, simple to understand, highly practical.

The series 1+2+3... on the other hand is very much ill behaved. There are other bad series which still behave better than this one. For example +1-1+1-1... where the partial sums oscillate between zero and one. We could just take the average and get 1/2. While in such a case it seems reasonable to call the result a sum, it really is already something new:

1+2+3 ... n –> (1+2+3...n) / n

An oscillating series like +1-1+1-1... can be handled by Cesaro summation, the fancy name for a family of methods similar to averaging. Instead of simply dividing by n one can imagine dividing by something larger, to capture faster growing, ill behaved series.

1+2+3 ... n –> (1+2+3...n) / e^(q·n)

Abel summation amounts to dividing by an exponential function instead. Since that is a smooth function one can retain some smoothness by introducing an additional parameter, we can then again take a limit over. Roughly, let q go to zero like this:

With that we can find an answer to the hypergeometric series:

1-1+2-6+24-120...

...whose elements are just the factorial n! peppered alternatingly with minus signs. For it, Euler derived the value 0.5963473623... following half a dozen distinct ways! It's also interesting to note that Abel summation is consistent with Cesar summation, whenever the latter is defined, the former gives the same answer.

Does that mean it's hard to come up with an method to arrive at any number? No, that's actually too easy. We're looking for "interesting" methods, ones that obey "interesting" rules. Like these:

regular: Yields the known result when applied to ordinary convergent series
linear: A(k·r + s) = k·A(r) + A(s)
stable: Leaving off initial elements gives the same result.

A regular method is interesting because it can then be seen as a generalization of the common Cauchy series. Linearity is just a nice property also obeyed by them. As is stability. The Abel sum is all of these.

The word sum is already a bit of a stretch. Okay, classic Cauchy series are still used, and only the terms change: something gets multiplied to them. And what about that free parameter?

If you visualize the partial sums 1+2+3... as a step function, you could put a parabola through the midpoints of the steps (look at the image now). A parabola feels right because the partial sum formula for our triangle numbers involves some x^2. Here's the thing: That parabola intersects the y-axis at y = -1/12!

The curve is some kind of smoothed version of our series. Initially defined only for integers, we can provide intermediate values by distorting our formula with the additional parmeter.

Smooth, like in analytic continuation! If we only could approach our series from another angle! So let's whack complext numbers through our formulas, they got an angle, eh? If it's smooth enough, we can extend our mutated series to the whole complex plane, and can then indeed approach our problematic value from another angle! It's called Euler summation, so he was there.

Meet zeta regularization, a nonlinear method that, of course, also gets us to -1/12:

1+2+3 ... n -> 1/1^s + 1/2^s + 1/3^s ...

When that one converges for some s, and can from there be analytically continued to s = -1, the value obtained is called the zeta regularized sum of the original series.

To learn about a related concept with prominence in physice, and often confused with zeta regularization, you may want to look up Dirichlet series.

Interestingly, all this connects Euler's -1/12 to the zeta function. He also liked to change the order of the elements in the series, and for that one may want to require

finite reindexability: An alternative to stability

Puzzle: I'd like to know why i shouldn't confuse regularization with renormalization. The former i just told you about, the latter is required to keep the sums associated to Feynman diagrams from running off to infinity.

For nitpickers: Those aren't sums! You could still write them as sums or otherwise pretend they were, but you'd be missing a big chunk of beautiful mathematics...

references

Here's my post with the preceding discussion "Diagram 20: X-Rays of the zeta function"

Here's a very nice, new and free book by Barry Mazur, William Stein: "Primes / What is the Riemann Hypothesis"
http://wstein.org/rh/

"Divergent series", a classic book by Godfrey Harold Hardy, a friend of Srinivasa Ramanujan!
https://archive.org/details/DivergentSeries

has a nice post about -1/12 here:

Another post by , "Prime numbers and the Riemann zeta function", a followup to 's post i linked last time:

As always with him, don't miss the comments!

took it up as part of a series on Riemann's hypothesis. It attracted quite some fuss:
ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = -1/12

has collected a bulk of links of reactions to the above video:
http://aperiodical.com/2014/01/an-infinite-series-of-blog-posts-which-sums-to-minus-a-twelfth/

In that one's main post, a link to Phil Slate, trying to put it on firm ground in a single blog entry: