1 + 2 + 3 + .. = -1/12

No, the math isn't sloppy, at least not in the way you imply. B_n are the Bernoulli numbers - I can't find a K(whatever) on the page. If you'd let me know which bits you don't get maybe I could help.

You can't rearrange a divergent series. It's a sin :-P. http://www.math.ucla.edu/~ralston/pub/Rearrange.pdf

I hope you get to see this, as I've trawled your web site and not found any way to get in touch. Based on what you've said here you're exactly the sort of person I'd like to have read through some stuff I'm writing. You only need to read as much as you're interested in, so I would hope we'd both get something out of it.

My email is in my profile if you'd like to get in touch. Obviously no obligation, so even a ping to say "no thanks" would be welcome.

Thanks.

> I made it through multivariable calculus and grads and so forth, but the computer scientist in me gets upset when I run into undefined or poorly specified notation.

Well, at the top of the 'Summation' heading it does mention Bernoulli numbers.

There's also a video on this explaining it more gently - submitted here:

https://news.ycombinator.com/item?id=7038809

In part it can be used to illustrate what bad things can happen when you do things that are intuitively reasonable. It's worth noting that Euler did exactly this kind of stuff when he solved the famous Basel Problem[0] of sum(1/n^2) giving pi^2/6. His manipulations turned out to give the right answer, these seemingly similar manipulations give "obvious nonsense."

Calculus is founded on doing odd things with infinite collections, and knowing for sure what works and what doesn't is important. When you do engineering you're pretty much guaranteed that "obviously right" things are actually right, but math allows us to explore places where our intuition is misleading, and helps prevent us from making mistakes.

And sometimes it's just fun. Have you ever written a Quine[1] program? Does everything have to have obvious and direct uses?

[0] http://en.wikipedia.org/wiki/Basel_problem

[1] http://en.wikipedia.org/wiki/Quine_%28computing%29

Watch this entertaining Numberphile video about this topic:

http://www.youtube.com/watch?v=w-I6XTVZXww

Apparently it's used in many areas in Physics.

Ah, mathematics. Most of it is absolutely useless, and yet, Physics and engineering won't stop finding new applications for the most abstract mathematical concepts. Modern physics would have been impossible to develop without the useless math concepts invented throughout the 19th century.

> useful in the study of divergent infinite series

The mathematics of infinite series in general has many existing applications outside of pure math, for example in physics. That said, I don't know if this theory in particular is applicable outside of theoretical mathematics (yet).

EDIT: It would actually be nice if the Wikipedia article would mention existing applications outside of mathematics - I think maths has a worse reputation in usefulness than it deserves because its applications are not highlighted enough.

If you look at the second comment on this question here: http://physics.stackexchange.com/q/5207/34235 it touches on the explanation as to why bosonic string theory has 26 dimensions (as a result of this sum of natural numbers resulting in -1/12)

To remind you to be careful when doing any kind of series expansion. It's more important than it sounds.

I don't know of any use behind the calculation of the value for this particular series. But Ramanujan summation (and other summation methods for divergent series) show up in the renormalization calculations in field theories. So this might sound like a random combination of fancy sounding words,but to give an idea of the connection: imagine you have some kind of integral that you want to evaluate or ODE you want to solve and you have some expansion that turns it into a series. But then you notice that a naive grouping of the terms in the series makes the sum go to infinity. But there are other ways where you can cleverly rearrange the grouping of terms so that it converges. So choosing a method for which this works (and gives a results that makes sense physically or agrees with other known exact results) is a serious business.

The "art of summing up divergent series" is a well established technique in quantum field theory (not only in string theory). Briefly, the underlying mathematical description of quantum field theory produces singularities. The latter are not physical (so their existence is due to our poor mathematical abilities) but they still contain useful physical information. This information can be extracted by "correctly" performing divergent sums. And its results are verified by experiment, e.g. in higher order corrections to particle masses.