I maintain that the sum grows without bound.

Rearranging an infinite series to create two series is not a valid operation. It leads to paradoxes. I remember pseudo proofs resulting in paradoxical results due to such rearrangements.

Consider the following infinite series, outside the context of this thread.

9 + 9 + 9 . . .

It is clearly unbounded and it is clearly the same as the following unbounded series.

[10 - 1] + [10 -1] + [10 -1] . . .

I do not think that rearranging to create the following is valid.

[10 + 10 + 10 . . .] - [1 + 1 + 1 . . .]

The latter becomes the difference of two unbounded series, creating a bit of confusion, and suggesting that the sum might be zero, which is a silly result considering the original series.