Can Infinite Sums Be Manipulated by Shifting Terms?

[jackferry]

I was reading the Wikipedia article about the sum 1+2+3+4+..., and I saw this explanation:
c = 1+2+3+4+5+6+...
4c = _4__+8__+12+...
-3c = 1-2+3-4+5-6+...
link: http://en.wikipedia.org/wiki/1_+_2_+_3_+_4_+_⋯
My question, as one who hasn't worked with infinite sums:
Why are you allowed to shift the numbers when adding/subtracting/manipulating infinite series. For instance:
b = 1+1+1+...
b = __1+1+...
thus b-b = 0 = 1
If shifting numbers is allowed, why can something like that be accounted for? Is it a dividing by zero, "dont touch that" kind of thing or is shifting series while manipulating them only allowed for certain series?
Also on Wikipedia (link: http://en.wikipedia.org/wiki/1_+_1_+_1_+_1_+_⋯), I saw that the sum of 1+1+1+... = -1/2. If you add an infinite number of 1+1+1+... together after shifting them, you can make the original 1+2+3+4+...
Here is what I am saying:
b = 1+1+1+1+1+...
b = __1+1+1+1+...
b = ____1+1+1+...
and so on...
So if 1+1+1+... = b, b = -1/2, b+b+b+... = 1+2+3+4+... and 1+2+3+4+... = -1/12 does (-1/2)+(-1/2)+(-1/2)+... = -1/12?
Answers to those questions would be tremendously appreciated, as well as any critiques of my misunderstanding of this subject. Thank you for your time.
Bonus question: Has anyone figured out how an infinite sum of positive numbers equals a negative number? I'm not asking for proofs of the sum, just an explanation of this weird result.
p.s. Sorry for the underscores, I had trouble with the formatting.

 

[micromass]

Infinite sums (called series) are a very tricky subject. There are many possible definitions that one can follow in order to calculate a series.

So, what does the series equal in general? In order to know that, one need to know about "convergence of sequences". This is crucial, but I won't go into it in detail. Let's give an example, let's take

1,~\frac{1}{2},~\frac{1}{4},~\frac{1}{8},...

we see that every term in the sequence goes closer and closer to $0$ (while perhaps never reaching it fully). We say that the sequence converges to $0$.

Now, when given an infinite sum $x_1 + x_2 + x_3 + x_4 + ...$, we look at the sequence

x_1,~x_1+x_2,~x_1+x_2+x_3,~...

and we see whether that gets very close to a certain number. In this case, we have

1,~1+2,~1+2+3,~1+2+3+4,~...

we see easily that this sequence grows arbitrarily large. We say that the series goes to infinity. That is

1+2+3+4+... = +\infty

In the same way, we see that

1+1+1+1+... = + \infty

The first thing you notices whas about subtracting infinite series, so let's do this. You want to know about

(1+1+1+1+...) - (0+1+1+1+1+...)

But we see that this is actually equal to $+\infty - \infty$ and it is a convention in mathematics that is undefined. So the difference does not exist. We do this to avoid nonsense results.

Now, wait up. I just told you that

1+2+3+4+5+... = +\infty

But you have no doubt seen many sources that it equals $-1/12$. Which is it?? Well, the answer is both. My answer of $+\infty$ is the answer under the usual definition of convergence (the one I explained about). But there are many different definitions of convergence out there. A popular alternative definition is Ramanujan summation (http://en.wikipedia.org/wiki/Ramanujan_summation). Under this definition, we get a sum of $-1/12$.

The two notions of convergence are obviously not equivalent. And both are useful facts.

Finally, there are many "math populizers" who try to amaze the public with impossible things like

1+2+3+4+... = -\frac{1}{12}

I have very mixed feelings about this. It's obviously a good thing to get people interested in math. But I don't think it's ok to "lie" to the public and omit what exactly we mean with an infinite sum. The identity is presented as some kind of amazing mindblowing result, while it really isn't.

 

[jackferry]

Thank you very much, this makes much more sense. I was getting very confused and a little annoyed about the whole 1+2+3+4+... = -1/12 thing, but it helps a lot to have some context.

 

[micromass]

Thank you very much, this makes much more sense. I was getting very confused and a little annoyed about the whole 1+2+3+4+... = -1/12 thing, but it helps a lot to have some context.

One of the things that mathematicians keep themselves busy with is to take a question with an undefined/infinite answer, and to see if we can re-interpret the question to give some other answer.

Here are other examples of this phenomenon:
Obviously, we have
1+2+4+... = +\infty
But we can also argue as follows:
If $S=1+2+4+...$, then $S-1 = 2(1+2+4+...) = 2S$. And if we solve $S-1 = 2S$, then we get $S=-1$. This equality is actually true if we interpret convergence as convergence in the $2$-adic norm.

Another one is $\infty ! = \sqrt{2\pi}$.
Or this: http://link.springer.com/article/10.1007/s00220-007-0350-z

All these things make sense and are true, but only if we interpret the question and definitions in the right way. This interpretation is usually not the most intuitive one, but it nevertheless does find applications.