Comparing divergent series - how to interpret the relationship between these two series?

Question

Say we have the series $1+2+4+...$ and the series $2+4+8+...$

It seems that the first series is half of the second series, but looking from another point of view it seems that the first series is actually the second series plus one.

How would we compare the "values" of divergent series such as these?

EDIT: I guess some explanation is due for why I asked this wacky question. I was looking at questions from my HW which ask me to compute sequences that are gotten as a result of "shifting" - i.e. deleting the first $n$ terms in a sequence so that everything is moved $n$ places to the left. The confusion arose when, upon observing that the value of the $k$th term of a shifted geometric sequence is some factor of the $k$th term of the original sequence, and that it also seems like we are "taking away" from the original sequence at the same time, I confused the relationship of the $sums$ of the sequences with the relationships of the individual terms in the $n$th spots of the respective sequences. But of course, thinking this way didn't seem to make sense, which is why I asked this question - and then found out that I was right insofar as it didn't make sense, and thus discovered my mistake.

Answer 1

To compare the divergent series you can compare their asymptotic behaviour (of the first $n$ terms) \begin{align} 1+2+4+\ldots+2^{n-1}&=2^n-1,\\ 2+4+8+\ldots+2^n\ \ \ &=2(2^n-1). \end{align} In this sense, "the first is the half of the second" is more preferable comparison than "plus one".

Answer 2

Both explanations are correct. Neither converge. Both diverge to positive infinity. What more interpretation do you need? There's no "values" to compare as both diverge.

Answer 3

What you have observed is colloquially written as "$\infty/2 = \infty = 2\infty$", which describes a phenomenon that is generally accepted as true. See Hilbert's hotel.