1+2+3+4... = -1/12

[Reincarnate]

check out dirichlet series and grandi series

this also reminds me: check out thompson's paradox. edit: http://en.wikipedia.org/wiki/Thomson's_lamp

It's really easy to abuse the concept of infinity

[reuben_tate]

Zapmeister...you can't use (1-x)^(-2) at x=1 as an analytic continuation of 1+2x+3x^2+... because (1-x)^(-2) itself is not analytic at x=1. You have to remember that analytic continuations are unique (in some sense). I still don't see any contradictions or inconsistencies when we use the notion of analytic continuation to "redefine" things.

[stargroup100]

Originally Posted by Zapmeister

you'd never write it out as 1+2+3+4+... because nobody knows what you mean when you're writing down the sum of a divergent series.

I agree.

Originally Posted by stargroup100

The reason why this result can even make sense is because we're talking about different contexts. Mathematical structures in different domains have different properties. When you have problems in which multiple answers can be justified, the conditions for the problem must be explained to obtain a particular solution.

The condition for this problem is talking about the riemann-zeta function (in a nutshell).

[reuben_tate]

I think the best way to explain this is by looking at an analog of a much more simpler example. When it comes to mathematics, we like to define a lot of things. However, sometimes we're stuck by the definition we have which restricts us. So we look for a way to "extend" our definitions to:
1) make everything still consistent with the old definition (i.e. not break maths)
2) have "nice" features in this new definition that is desirable

Let's take an example with exponentiation. When first introduced to exponentiation, we learn this as repeated multiplication: if we see something like 2^3, it's simple: 2^3 = 2*2*2 = 8. However, then you see something like 4^(3/2) and you're like "da fuq? how am I supposed to do one and a half multiplications?" With our original definition, we are powerless, we can not do anything, we are dumbfounded with the notion of trying to 1.5 multiplications. However, we can redefine exponentiation for fractional powers in a way that is still consistent with the previous rules and we find that 4^(3/2) = sqrt(4)^3 = 2^3 = 2*2*2 = 8.

We had to redefine what exponentiation meant for fractional powers to solve the problem. Did we redefine in any ol' way? No, we had to redefine it in a way that was still consistent with the previous rules. Think of this as the "fractional continuation" of the exponentiation function. We could have left it "as is" and just say it means nothing when our power isn't a whole number, but we wanted to work beyond the whole numbers. Mathematicians have extended this definition all the way up to the real numbers.

What about complex numbers? Well, looking only at the definition for what we have for real numbers, we can't do anything. We can redefine exponentiation for complex numbers quite nicely in a way that makes the function f(z) = z^c analytic, where z and c are complex numbers. What it means for a function to be analytic is beyond the scope of this post, but it's basically a "nice" property that is desirable for a function with complex variables. Although technically, this isn't an analytic continuation of f from the reals to the complex, the idea is nearly the same: we can extend the domain of many complex functions using analytic continuation (and since it can be shown that analytic continuation is unique, we have less to worry about in terms of inconsistencies!)

So although 1+2+3+4... doesn't really equal anything (unless you count infinity) with our original definition (i.e. the value of the limit of the partial sums), we can extend our definition by means of analytic continuation and result in -1/12.

So is 1+2+3+4...=-1/12? Both yes and no. Sticking with the usual definition which most of us are familiar with, this is bogus nonsense. Using this other definition which some mathematicians (particularly complex analysts and it seems quantum physicists use): yes. In this re-definition correct? Definitions are neither correct or incorrect, they just are so whether this re-definition using analytic continuation is a good one...that's your call.

[igotrhythm]

Originally Posted by reuben_tate

barf

Excuse me while I pick up the pieces of my brain that are all over the floor.

[Riotpolice]

This thread has got to be the flagship of the critical thinking thread.

[stargroup100]

which is funny because there's really not much debate in this thread

[Zapmeister]

Originally Posted by reuben_tate

Zapmeister...you can't use (1-x)^(-2) at x=1 as an analytic continuation of 1+2x+3x^2+... because (1-x)^(-2) itself is not analytic at x=1.

crap... you got me. you got me damn good.

best i can say now is that if you try to define sums of divergent series using the formulae and stuff you derive from convergent series, you have to throw away a lot of favourable nice assumptions you make about how they behave.

e.g.

sum(n: n=1 to infinity) = x (here x is assumed to be -1/12 but that doesn't matter)
=> sum(n: n=0 to infinity) = x
=> sum(n-1: n=1 to infinity) = x
=> sum(n: n=0 to infinity) - sum(n-1: n=1 to infinity) = 0
=> sum(1: n=1 to infinity) = 0
=> sum(1: n=0 to infinity) = 0
=> sum(1: n=0 to infinity) - sum(1: n=1 to infinity) = 0
=> 1=0

however i'm pretty sure term-by-term addition/subtraction like this is not justified in the way stargroup100 is trying to treat these things, although he never explicitly states that in any of his posts.

i'm still trying to find a contradiction that i'm satisfied with but for the moment it looks like you got me here >_<

edit: i was hoping my 300th post would be less self-demeaning than this but whatever

[rushyrulz]

I'd have to say, the most intriguing mathematical truth is:

[stargroup100]

Originally Posted by Zapmeister

however i'm pretty sure term-by-term addition/subtraction like this is not justified in the way stargroup100 is trying to treat these things, although he never explicitly states that in any of his posts.

you're right, I didn't explicitly state anything of the sort. because I agree with you.

[jaz_pup]

I'm (a bit) surprised so many people seem to have taken complex analysis.


I find it interesting that so many people get so heated over what they believe is the right answer regarding math. It's like that one problem floating around involving the order of operations...I'm pretty sure there have been like legit fist fights over that one...probably.

[21992]

Funny we were just discussing this in my calculus class. In fact the infinite series of positive numbers is divergent (infinity or undefined). This instance is kind of like how my calc teacher told us that taking the integral of natural log yielded 1=0 . But then there came the +C that corrected the equation. I think something like that will come up but for now it can't be explained. I find it funny that the tool we created to better understand our universe, math, we can't even fully figure out ourselves.

[21992]

A quick google search https://www.youtube.com/watch?v=w-I6XTVZXww

[reuben_tate]

Originally Posted by Zapmeister

e.g.

sum(n: n=1 to infinity) = x (here x is assumed to be -1/12 but that doesn't matter)
=> sum(n: n=0 to infinity) = x
=> sum(n-1: n=1 to infinity) = x
=> sum(n: n=0 to infinity) - sum(n-1: n=1 to infinity) = 0
=> sum(1: n=1 to infinity) = 0
=> sum(1: n=0 to infinity) = 0
=> sum(1: n=0 to infinity) - sum(1: n=1 to infinity) = 0
=> 1=0

There's a lot of assumptions being made here and a lot of things done without justification. The terms in an infinite series are like the flesh of a cacti...they don't want to be ****ed with.

However, there is a lesson to be learned here. The lesson being that this way of defining how to given values to divergent series isn't "nice" and doesn't have the "nice" properties like the ones we see with convergent series (see this and this). We know with convergent series we can add term-by-term...but your thing shows that that is clearly not the case anymore (or else otherwise we get the absurdity that 0=1).

To be honest...this way of defining how to assign values to divergent series doesn't sit too well me either. It works well for complex analysts since analytic functions are "nice". However, this definition doesn't have a lot of the "nice" properties one is looking so if you find a "nicer" one let me know

[stargroup100]

Originally Posted by jaz_pup

I find it interesting that so many people get so heated over what they believe is the right answer regarding math.

I don't really care about what people believe is the right answer. Whether or not they believe doesn't make the answer right or wrong.

What I do care about however, is how people approach and understand math. There are way too many misconceptions about the nature of mathematics and how it's applied.

Example of what I mean:

Originally Posted by 21992

I think something like that will come up but for now it can't be explained. I find it funny that the tool we created to better understand our universe, math, we can't even fully figure out ourselves.

"it can't be explained"
Actually this topic has been explored pretty well. It's nothing new for mathematicians, and while there are questions that are still unanswered, we generally have a pretty good idea of what goes on here.

Math can be thought of as a tool we created to better understand our universe, this is fine. There should be absolutely nothing surprising about not being able to fully figure out mathematics as a whole. Just because you created something doesn't necessarily mean you understand it completely, and there are countless examples of this in the real world. Why is math special in that it needs to be completely understood?

Mathematics in many ways has infinite complexity in many dimensions. Not only are there probably an infinite number of systems and objects, but different mathematical objects behave differently under different circumstances/domains/etc. There are no "concrete rules" in mathematics. You absolutely must follow the rules you set up within your own system, but none of these rules are universal to all systems. In fact, complete axiomization of all mathematics has already been proved to be impossible (See Gödel's incompleteness theorems).

What is important is not whether or not we fully understand math, it's how much we understand and how we use it. Over time we build upon our existing knowledge and expand it, which then results in more applications to real life. There are always going to be things we don't know; it's literally impossible to know everything. Our objective is not to have perfect knowledge, but to increase our knowledge as much as possible.

[jaz_pup]

I think this problem is too advanced and we should find something fascinating in math that's in the realm of understanding for everyone. Any ideas?

[stargroup100]

Originally Posted by jaz_pup

I think this problem is too advanced and we should find something fascinating in math that's in the realm of understanding for everyone. Any ideas?

The details surrounding this result is quite advanced, yes, but that doesn't mean you shouldn't find it fascinating. You really only need to know one thing to appreciate this fact: In some contexts this result is justified. You can go one step further: You can use analytic continuation to expand the domain of a function. One step further: This result is related to the behavior of the riemann-zeta function. You can go as far as you like, but the first statement alone allows us to appreciate the surprising result.

When you find something fascinating, it doesn't have to be something you can understand. There are lots of facts about the world, not just mathematics, that are amazing, and many of them even top academics don't understand yet.

It's more important to first understand mathematics as a whole, the things I've mentioned in my last post. Once people understand these basic ideas, there will be far fewer misunderstandings about math, far more people that will appreciate it, and it will contribute greatly to people being better at math.

[Zapmeister]

i'm not seeing it. why should/would i appreciate the result "1+2+3+4+stuff = -1/12", or find it "fascinating", if by analytic continuation any series that has a rule for the terms can be "summed" by taking a power series or (in your case here) a dirichlet series and extending it outside its radius of convergence. if it's about the applications of the actual value -1/12 then why bother with writing it out like that instead of writing it as zeta(-1) and using properties of the zeta function.

have at me.

[jaz_pup]

Originally Posted by stargroup100

The details surrounding this result is quite advanced, yes, but that doesn't mean you shouldn't find it fascinating. You really only need to know one thing to appreciate this fact: In some contexts this result is justified. You can go one step further: You can use analytic continuation to expand the domain of a function. One step further: This result is related to the behavior of the riemann-zeta function. You can go as far as you like, but the first statement alone allows us to appreciate the surprising result.

When you find something fascinating, it doesn't have to be something you can understand. There are lots of facts about the world, not just mathematics, that are amazing, and many of them even top academics don't understand yet.

It's more important to first understand mathematics as a whole, the things I've mentioned in my last post. Once people understand these basic ideas, there will be far fewer misunderstandings about math, far more people that will appreciate it, and it will contribute greatly to people being better at math.

I never said it wasn't fascinating, I just said we should discuss something more easily understandable that is fascinating. And I think I understand enough math, I gave the suggestion so that we could all collectively have a more useful discussion since it might be something more accessible as not many people have studied (as far as I know) the Riemann-Zeta function.

[stargroup100]

-.-

Originally Posted by Zapmeister

by analytic continuation any series that has a rule for the terms can be "summed" by taking a power series or (in your case here) a dirichlet series and extending it outside its radius of convergence.

if you learn this fact first then maybe it's not so fascinating. someone that doesn't know this could be fascinated by the fact that that sum can be justified.

when I first learned about calculus, the idea that you can find the area under any polynomial curve in trivial time was mindblowing to me. now I take it for granted because I'm so familiar with basic integrals, but that didn't make it any less surprising when I first learned about it

Originally Posted by Zapmeister

if it's about the applications of the actual value -1/12 then why bother with writing it out like that instead of writing it as zeta(-1) and using properties of the zeta function.

because a layman wouldn't understand anything about the zeta function unless you used layman terms.

Originally Posted by jaz_pup

stuff

ah I misunderstood you. my bad.