[cohomologo]

There is a much simpler mathematical treatment of this series that appeared in my class on quantum field theory in the computation of the vacuum energy. In that case, the series was treated as

lim \epsilon -> 0+ ( \sum_{n=1}^\infty n e^{- \epsilon n} + ...),

that is the series was multiplied by a decaying exponential function with a rate of decay that goes to zero. This sum can easily be evaluated for small epsilon takes the form

sum = 1/epsilon - 1/12 + ...

Crucially, there was another term in the calculation that naturally appeared that canceled the 1/epsilon. Without that other term, the sum would of course be infinite when epsilon -> 0.

This is much simpler than analytic continuation through the complex plane, and again, this is how the physics calculation appears in QFT courses. There is no need to appeal to complex analysis here, which leads to all of this mysticism and confusion.

[archgoon]

That's completely unreadable.

So how exactly is it that by multiplying a regular sum of positive numbers by a decaying exponent do you get a negative number when you take the limit?

[bzbarsky]

I think the argument goes like this:

1) You introduce a family of series, parametrized by epsilon, whose terms, for small epsilon, closely approximate your original series. Sort of; obviously for large n they don't. But the idea is that if you pick any N and delta you can pick epsilon such that for n < N the approximation is within delta of the actual terms.

2) You show that the series in this family can all be summed and the sum of each one is 1/epsilon - 1/12 + O(epsilon).

Now of course what this means is that the sums blow up as you approximate your original series better and better, since 1/epsilon gets large. That's good, because your original series totally diverges off to infinity. ;)

The part after this point I'm less clear on, but it sounds like in the computation involved what you actually have is your (divergent) series 1+2+3+... plus some _other_ (also divergent, going off to negative infinity) stuff. And that you might be able to arrange things such that the other divergent stuff looks like -1/epsilon, cancels out the 1/epsilon from your approximation, and you come out with the sum of the two things being -1/12.

The obvious issue here is that once you start adding up divergent things by rearranging terms and telescoping you can come up with whatever answer you want: see <https://en.wikipedia.org/wiki/Riemann_series_theorem>. So this procedure all only makes sense if there are some sort of fundamental reasons to think that this particular rearrangement is the "right" one in some sense.

[archgoon]

Thank you for clarifying the approach (and with not trying to use latex!) :).

The issue is, of course, the illegitimate manipulation of a diverging series, which was the exact issue that prompted the original article (due to Numberphile doing it) in the first place.

[bzbarsky]

One thing I _will_ say for the "decaying exponent" thing is that it's somewhat similar in concept, but only somewhat, to the "smoothed sums" thing described in <https://terrytao.wordpress.com/2010/04/10/the-euler-maclauri.... That is, if you use "e^(-x)" as your "cutoff function" and set epsilon to 1/N the two start looking quite similar.

Now e^(-x) is a totally bogus "cutoff function" per the definition in Terry's blog post, since it is not compactly supported, but it _is_ bounded, _does_ equal 1 at 0, and drops off fast enough that for practical purposes it can be used to do smoothed sums. In particular the smoothed sums will converge for most cases (e.g. anything where the sequence we're "summing" has at most polynomial growth will do so), which means you can at least try to do the rest of the analysis. I suspect, but have not checked, that the other places where compact support is used in his presentation also work out for the sorts of sequences we're talking about.

Either way, the upshot is that in some sense you have some sequence of approximations to your "actual" sum, indexed by N, and you show that for large N they all look like "power series" in 1/N which allows some finite number of negative exponents and all the approximations have matching coefficients for the negative exponents and the same constant term. And then you compute that constant term. Calling that the sum of the series is nonsense, of course, but it can still give you interesting information about something, maybe.

[tamana]

You don't. You get -1/12 plus a positive infinite expression, which is then reduced by a positive infinite expression later. Its theoretical physics, not math.

[archgoon]

> which leads to all of this mysticism and confusion.

Funny, I thought the entire point of this 'simplification' was not to be mystical and hand wavey.