Adding sum of negative integers to the sum of positive integers - how would value look like?

Question

I know that the sum of positive integers will diverge and that the sum of negative integers will diverge. I also know that "the sum of all integers" will likely be meaningless (it depends on the order of sequence).

But let us consider the following case:

Let us first add all positive integers in order from 1 to forever. Then add all negative integers in order from -1 to -infinity. Then what would sum be? Would this be indeterminate?

Answer 1

In general, when you've got a divergent series, by reordering the terms, you can make it appear to converge to almost anything. And when you say "Let's us first add the positive integers from 1 to forever," you've started off by saying "Let's compute a limit that does not exist." Anything else you say after that is pretty much meaningless from the point of view of formal mathematics.

Sorry to be disappointing here, but that's the way it is.

As an exercise, you might consider this: the sum $1 + (1/2) + (1/3) + ... $ diverges. $1 - (1/2) + (1/3) - (1/4) + ... $ converges. In fact, you can show that by placing "+" and "-" signs in the right places, you can arrange for these terms to sum up to any positive irrational number. (The algorithm is pretty easy: if your number is $r$, sum up terms until you get larger than r; then subtract terms until you're less than $r$. Then add terms until you're greater, and so on. Proving convergence takes a little work. :)

Answer 2

There are several ways to interpret your question:

1. "Let us start adding all positive integers starting from 1 and going forever. Then - -" Alas, if we first do something forewer, we cannot do anything after that, so this is not defined at all.

2. "The sum of positive integers plus the sum of negative integers." In order for this to be meaningful, you need to extend the set of numbers to contain $\infty$. However, this becomes $\infty + (-\infty)$ which is not determined.

3. "Start adding negative integers to the sum of positive integers in order." Again, we need to extend the set of numbers to contain $\infty$. Now, this may of example be interpreted as the limit of the series $\sum_{k=0}^\infty a_k$ where $a_0=\infty$ and $a_k=-k$ for $k>0$. Every partial sum $\sum_{k=0}^n a_k$ is $\infty$ so the limit of this series is $\infty$.

4. I guess there are plenty of other ways, too.