TITLE: Sum of all integers
QUESTION [32 upvotes]: 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?

REPLY [2 votes]: It is generally considered that the series
$$ 1 - 1 + 1 - 1 + 1 - 1 \pm \cdots$$
does not converge, since the partial sums alternate between $1$ and $0$
forever.
Even if you suppose that it's OK to rewrite a double-ended sum by taking
terms alternately in each direction,
you do not get a better result by summing
$$ 0 + 1 - 1 + 2 - 2 + 3 - 3 + 4 - 4 \pm \cdots.$$
You could put parentheses around pairs of terms and pretend
that both of these series are the same as
$ 0 + 0 + 0 + \cdots,$
but they are not.
You can try rearranging the positive and negative integers in
any sequence you like; none of it is going to stop the series from
eventually taking larger and larger steps up or down with each new
partial sum.