In this thread someone points out you seemingly can derive the equation 1+2+4+8+16+… = -1 as follows:

x = 1+2+4+8+16+…

x = 1+2(1+2+4+8+…)

x = 1+2x

x = -1

My question is, what do mathematicians take to be the right way to respond to this.

Is there a consistent way to build a mathematics of such “sums”? Or do there turn out to be more than one way to “sum” some infinite series? If the latter, then do we rule out such sums on this basis, or come up with a math which takes into account not only the members of the list of addends but also their order and groupings?

Or do we just call these kinds of sums “undefined” and move on?

It appears to me that there are some infinite sums which do not come out to a nice neat answer like this one does (I think 1+2+3+4+5… is an example of this), while some (like this one) do. Am I right to think this is the case? Or is there some way to find such a “sum” for *any* arbitrary infinite list of addends?

And so on.

-FrL-