1 + 2 + 3 + .. = -1/12(en.wikipedia.org)
en.wikipedia.org
1 + 2 + 3 + .. = -1/12
http://en.wikipedia.org/wiki/Ramanujan_summation
4 comments
The title is fairly misleading. The summation is not equal to -1/12, it is -1/12(ℛ), where the (ℛ) indicates that the answer is actually infinite, but it has a defined relationship to other series denoted in the same way.
Is the math here sloppy, or is it just me? Third equation they're bringing out B(i) and K(whatever).
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.
"Oh," someone will reply, "You didn't know that B(i) is the Frogglegoop function for ..." and I'll say, "Nope."
I rage quit denotational semantics after failing to find definitions of the hieroglyphs in two different textbooks. If you're going to bring greek and single letter function names to the game, you could at least provide a symbol table, eh?
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.
"Oh," someone will reply, "You didn't know that B(i) is the Frogglegoop function for ..." and I'll say, "Nope."
I rage quit denotational semantics after failing to find definitions of the hieroglyphs in two different textbooks. If you're going to bring greek and single letter function names to the game, you could at least provide a symbol table, eh?
You can't rearrange a divergent series. It's a sin :-P. http://www.math.ucla.edu/~ralston/pub/Rearrange.pdf
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.
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.
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.
Well, at the top of the 'Summation' heading it does mention Bernoulli numbers.
I might have missed this but what use does this have? (Honest question.)
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
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
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.
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.
> Modern physics would have been impossible to develop without the useless math concepts invented throughout the 19th century.
Case in point: imaginary numbers were "discovered" in the 16th century and were widely considered to be "useless" by contemporary mathematicians and scientists. However, in the 19th century it was realized that complex arithmetic is perfect for analyzing steady state alternating current circuits.
Case in point: imaginary numbers were "discovered" in the 16th century and were widely considered to be "useless" by contemporary mathematicians and scientists. However, in the 19th century it was realized that complex arithmetic is perfect for analyzing steady state alternating current circuits.
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)
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.
> 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.
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.
The Wikipedia article on the method used to calculate this sum mentions applications to quantum field theory [1]. Links to applications might be an interesting category of metadata for Wikipedia math articles which could be semi-automatically constructed.
[1] http://en.wikipedia.org/wiki/Ramanujan_summation
[1] http://en.wikipedia.org/wiki/Ramanujan_summation
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.
http://www.youtube.com/watch?v=w-I6XTVZXww
Apparently it's used in many areas in Physics.
I watched a lot of their videos, and until now they were pretty honest. This video is trying to be astonishing by explaining high level math to regular people without clarifying that holds true under different circumstances. That will make people confused over what the sum really is and feels deceptive, since they don't explain that if you are not in a special field of math/physics, the sum is divergent(inf).
To remind you to be careful when doing any kind of series expansion. It's more important than it sounds.