Questions About Infinite Sums

[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.

 

[CellCoree]

Dear reader,

Thank you for reaching out with your questions about infinite sums. I would like to provide some insights and answers to your inquiries.

Firstly, it is important to note that infinite sums, also known as infinite series, are a fundamental concept in mathematics and have been studied extensively by mathematicians for centuries. They are often used in various fields of science to model and describe natural phenomena, and have also been the subject of philosophical debates.

Now, to address your first question about shifting numbers in infinite sums. The concept of shifting numbers is allowed in infinite series because of the properties of convergence and absolute convergence. Convergence refers to the idea that an infinite series approaches a finite limit as the number of terms increases. Absolute convergence, on the other hand, refers to the convergence of an infinite series regardless of the order in which the terms are added. In other words, for an infinite series to be absolutely convergent, the rearrangement of terms should not affect the final sum.

In the case of the infinite series 1+2+3+4+..., it is not absolutely convergent. This means that rearranging the terms can result in different sums. This is why, when we shift the numbers in the series, we get different results. In your example, b-b=0 because the series is not absolutely convergent, and the shifting of terms affects the final sum. It is not a "don't touch that" kind of situation, but rather a result of the properties of infinite series.

Moving on to your second question about the sum of 1+1+1+... equaling -1/2, it is important to note that this result is not a "proof" or a definite answer. It is a result obtained by applying a mathematical concept called analytic continuation. This concept allows us to extend the definition of a mathematical function beyond its initial domain. In this case, the initial domain of the function is the set of all positive integers, but by using analytic continuation, we can extend the definition to include negative numbers and fractions as well. This is a common practice in mathematics, and it is not limited to infinite series.

To your bonus question about how an infinite sum of positive numbers can equal a negative number, the answer lies in the concept of analytic continuation and the properties of infinite series. As mentioned earlier, the initial domain of the function is extended to include negative numbers and fractions. This extension changes the behavior of the function