Sum of all integers

No, I'm not talking about $-\frac{1}{12}$.

I was talking with someone the other day, and they said that the sum of all integers, positive and negative, is zero because they all cancel each other out. Basically,

$$\cdots+(-3)+(-2)+(-1)+0+1+2+3+\cdots=0$$

I'm skeptical, given how infinity tends to never work the way I want it to, but I have no math to back this.

What I'm asking is this:

Is the sum of all integers 0?

Solution 1:

We model the concept of "infinite sum" by taking limits of finite sums. This model has the "flaw" that the value of an infinite sum will depend on the order we sum the terms (compared to finite sums, where the order doesn't matter). If you think the sum of all integers as

$$\lim_{n\to \infty} \sum_{i=0}^n (i-i)$$

then certainly the sum is equal to $0$, however, if you think this sum as

$$\lim_{n\to \infty} \left(\sum_{i=0}^n (i-(i+1))\right)$$

(All the integers appear as a sumand) then the sum is not zero!, The limit doesn't exist.

Solution 2:

Not really. It's not even a particularly well-posed question, to be honest. The problem is that most the usual suspects behind assigning values to divergent sums (like the statement that $1+2+3+\ldots=\frac{-1}{12}$) work on sums of the form: $$\sum_{i=1}^{\infty}s_i$$ for some sequence $s_i$. Why is this a problem? Well, they simply can't handle it if your series extends infinitely in both directions.

Okay, we could try to work around it by asking about $$0+1-1+2-2+3-3+4-4+\ldots$$ though I'm not sure what techniques might handle such a sum. However, the trouble is that we could equally well ask about: $$0+1+2-1+3+4-2+5+6-3+7+8-4+\ldots$$ where we include two positive terms for each negative one - and we'll probably get a different answer. So what order are we supposed to choose?

To illustrate the point, we can use the usual pseudo-algebraic techniques, or something more rigorous (like a Cesaro summation) to assign the value $\frac{1}2$ to the sum: $$0+1+0-1+0+1+0-1+0+1+0-1+0+1+0-1+\ldots$$ but, if we shuffle the zeros around to get the expression: $$1-1+0+0+1-1+0+0+1-1+0+0+1-1+\ldots$$ we end up assigning a different sum ($\frac{1}4$), despite the terms being the same. So we have some pathological behavior based on the ordering - which is troubling, because if we have no canonical order (i.e. we want to sum the set $\mathbb Z$, not some enumeration of it), we have no reason to think we can't order it so as to convince our method to spit out whatever number we want!

Sadly, the notion of summing a set is, almost exclusively, a measure theoretic concept applying to only absolutely convergent series - it doesn't even work on conditionally convergent series, let alone divergent ones. Worse, other techniques tend to want to examine a generating function, but they're not bidirectional unless we start talking about coefficients of $x^{-1}$ and so on (which we could do, but wouldn't be as useful as usual, since the main advantage of such a generating function is being able to work with the series near $0$, but we just plopped a singularity there...)

Solution 3:

Certainly $\sum\limits_{n = -\infty}^\infty n$ is divergent because for each positive integer $N$, $\sum\limits_{n = -2N}^{2N+1} n = 2N+1 \ge 2$.

It appears that the person you spoke with interpreted $\sum\limits_{n = -\infty}^\infty n$ as $\lim\limits_{n\to \infty} \sum\limits_{i = -n}^n i$, in which case the sum will be zero. However, this is incorrect. A bi-infinite series $\sum\limits_{n = -\infty}^\infty a_n$ converges if the double sequence

$$A_{m,n} := \sum\limits_{i = -m}^n a_i$$

converges. Note $m$ is independent of $n$. When $a_i = i$, we have $A_{2N,2N+1} \ge 2$ for all $N\in \Bbb N$, by the statement made above. This implies that the double sequence $A_{m,n}$ is unbounded, hence divergent. So $\sum\limits_{n = -\infty}^\infty n$ is divergent.

However, if you know a priori that a bi-infinite series $\sum\limits_{n = -\infty}^\infty a_n$ converges, then you can claim that the sum of the series is $\lim\limits_{n\to \infty}\sum\limits_{i = -n}^n a_i$.

Solution 4:

When you see written $\sum\limits_{i=1}^{\infty} x_i$ this term has a sense only if the series of $x_i$ is converging of diverging towards $\infty$ or $-\infty$. If you write $$ ... + (-3) + (-2) + (-1) + 0 + 1 + ... = 0$$

this sum is actually not defined. What would be the term of your sum? You could write $\sum\limits_{i=1}^{\infty} i$ + $\sum\limits_{i=1}^{\infty} -i$ which gives you $\infty - \infty$ but this is not defined.